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Metamath Proof Explorer

Theorem aceq0

Description: Equivalence of two versions of the Axiom of Choice. The proof uses neither AC nor the Axiom of Regularity. The right-hand side is our original ax-ac . (Contributed by NM, 5-Apr-2004)

Ref Expression
Assertion aceq0 ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑧 ∃! 𝑣 ∈ 𝑧 ∃ 𝑢 ∈ 𝑦 ( 𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢 ) ↔ ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑣 ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) )

Proof

Step Hyp Ref Expression
1 aceq1 ⊢ ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑧 ∃! 𝑣 ∈ 𝑧 ∃ 𝑢 ∈ 𝑦 ( 𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢 ) ↔ ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
2 equequ2 ⊢ ( 𝑣 = 𝑥 → ( 𝑢 = 𝑣 ↔ 𝑢 = 𝑥 ) )
3 2 bibi2d ⊢ ( 𝑣 = 𝑥 → ( ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
4 elequ2 ⊢ ( 𝑡 = 𝑥 → ( 𝑤 ∈ 𝑡 ↔ 𝑤 ∈ 𝑥 ) )
5 4 anbi2d ⊢ ( 𝑡 = 𝑥 → ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ↔ ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) )
6 elequ2 ⊢ ( 𝑡 = 𝑥 → ( 𝑢 ∈ 𝑡 ↔ 𝑢 ∈ 𝑥 ) )
7 elequ1 ⊢ ( 𝑡 = 𝑥 → ( 𝑡 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) )
8 6 7 anbi12d ⊢ ( 𝑡 = 𝑥 → ( ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ↔ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) )
9 5 8 anbi12d ⊢ ( 𝑡 = 𝑥 → ( ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ) )
10 9 cbvexvw ⊢ ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) )
11 10 bibi1i ⊢ ( ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ( ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) )
12 3 11 bitrdi ⊢ ( 𝑣 = 𝑥 → ( ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ( ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
13 12 albidv ⊢ ( 𝑣 = 𝑥 → ( ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∀ 𝑢 ( ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) ) )
14 elequ1 ⊢ ( 𝑢 = 𝑧 → ( 𝑢 ∈ 𝑤 ↔ 𝑧 ∈ 𝑤 ) )
15 14 anbi1d ⊢ ( 𝑢 = 𝑧 → ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ↔ ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ) )
16 elequ1 ⊢ ( 𝑢 = 𝑧 → ( 𝑢 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥 ) )
17 16 anbi1d ⊢ ( 𝑢 = 𝑧 → ( ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) )
18 15 17 anbi12d ⊢ ( 𝑢 = 𝑧 → ( ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ) )
19 18 exbidv ⊢ ( 𝑢 = 𝑧 → ( ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ) )
20 equequ1 ⊢ ( 𝑢 = 𝑧 → ( 𝑢 = 𝑥 ↔ 𝑧 = 𝑥 ) )
21 19 20 bibi12d ⊢ ( 𝑢 = 𝑧 → ( ( ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
22 21 cbvalvw ⊢ ( ∀ 𝑢 ( ∃ 𝑥 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑢 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑥 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
23 13 22 bitrdi ⊢ ( 𝑣 = 𝑥 → ( ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
24 23 cbvexvw ⊢ ( ∃ 𝑣 ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ↔ ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) )
25 24 imbi2i ⊢ ( ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑣 ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
26 25 2albii ⊢ ( ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑣 ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
27 26 exbii ⊢ ( ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑣 ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) ↔ ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑥 ∀ 𝑧 ( ∃ 𝑥 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) ) ↔ 𝑧 = 𝑥 ) ) )
28 1 27 bitr4i ⊢ ( ∃ 𝑦 ∀ 𝑧 ∈ 𝑥 ∀ 𝑤 ∈ 𝑧 ∃! 𝑣 ∈ 𝑧 ∃ 𝑢 ∈ 𝑦 ( 𝑧 ∈ 𝑢 ∧ 𝑣 ∈ 𝑢 ) ↔ ∃ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → ∃ 𝑣 ∀ 𝑢 ( ∃ 𝑡 ( ( 𝑢 ∈ 𝑤 ∧ 𝑤 ∈ 𝑡 ) ∧ ( 𝑢 ∈ 𝑡 ∧ 𝑡 ∈ 𝑦 ) ) ↔ 𝑢 = 𝑣 ) ) )