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Metamath Proof Explorer

Theorem axpweq

Description: Two equivalent ways to express the Power Set Axiom. Note that ax-pow is not used by the proof. When ax-pow is assumed and A is a set, both sides of the biconditional hold. In ZF, both sides hold if and only if A is a set (see pwexr ). (Contributed by NM, 22-Jun-2009)

Ref Expression
Assertion axpweq ( 𝒫 𝐴 ∈ V ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )

Proof

Step Hyp Ref Expression
1 pwidg ⊢ ( 𝒫 𝐴 ∈ V → 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 )
2 pweq ⊢ ( 𝑥 = 𝒫 𝐴 → 𝒫 𝑥 = 𝒫 𝒫 𝐴 )
3 2 eleq2d ⊢ ( 𝑥 = 𝒫 𝐴 → ( 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 ) )
4 3 spcegv ⊢ ( 𝒫 𝐴 ∈ V → ( 𝒫 𝐴 ∈ 𝒫 𝒫 𝐴 → ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 ) )
5 1 4 mpd ⊢ ( 𝒫 𝐴 ∈ V → ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 )
6 elex ⊢ ( 𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V )
7 6 exlimiv ⊢ ( ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 → 𝒫 𝐴 ∈ V )
8 5 7 impbii ⊢ ( 𝒫 𝐴 ∈ V ↔ ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 )
9 vex ⊢ 𝑥 ∈ V
10 9 elpw2 ⊢ ( 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ 𝒫 𝐴 ⊆ 𝑥 )
11 pwss ⊢ ( 𝒫 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ( 𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥 ) )
12 df-ss ⊢ ( 𝑦 ⊆ 𝐴 ↔ ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) )
13 12 imbi1i ⊢ ( ( 𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥 ) ↔ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
14 13 albii ⊢ ( ∀ 𝑦 ( 𝑦 ⊆ 𝐴 → 𝑦 ∈ 𝑥 ) ↔ ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
15 11 14 bitri ⊢ ( 𝒫 𝐴 ⊆ 𝑥 ↔ ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
16 10 15 bitri ⊢ ( 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
17 16 exbii ⊢ ( ∃ 𝑥 𝒫 𝐴 ∈ 𝒫 𝑥 ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )
18 8 17 bitri ⊢ ( 𝒫 𝐴 ∈ V ↔ ∃ 𝑥 ∀ 𝑦 ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝑥 ) )