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Metamath Proof Explorer

Theorem nalset

Description: No set contains all sets. Theorem 41 of Suppes p. 30. (Contributed by NM, 23-Aug-1993) Remove use of ax-12 and ax-13 . (Revised by BJ, 31-May-2019)

Ref Expression
Assertion nalset ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥

Proof

Step Hyp Ref Expression
1 alexn ⊢ ( ∀ 𝑥 ∃ 𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥 )
2 ax-sep ⊢ ∃ 𝑦 ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) )
3 elequ1 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑦 ↔ 𝑦 ∈ 𝑦 ) )
4 elequ1 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥 ) )
5 elequ1 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧 ) )
6 elequ2 ⊢ ( 𝑧 = 𝑦 → ( 𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝑦 ) )
7 5 6 bitrd ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ 𝑧 ↔ 𝑦 ∈ 𝑦 ) )
8 7 notbid ⊢ ( 𝑧 = 𝑦 → ( ¬ 𝑧 ∈ 𝑧 ↔ ¬ 𝑦 ∈ 𝑦 ) )
9 4 8 anbi12d ⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝑦 ) ) )
10 3 9 bibi12d ⊢ ( 𝑧 = 𝑦 → ( ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) ↔ ( 𝑦 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝑦 ) ) ) )
11 10 spvv ⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) → ( 𝑦 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝑦 ) ) )
12 pclem6 ⊢ ( ( 𝑦 ∈ 𝑦 ↔ ( 𝑦 ∈ 𝑥 ∧ ¬ 𝑦 ∈ 𝑦 ) ) → ¬ 𝑦 ∈ 𝑥 )
13 11 12 syl ⊢ ( ∀ 𝑧 ( 𝑧 ∈ 𝑦 ↔ ( 𝑧 ∈ 𝑥 ∧ ¬ 𝑧 ∈ 𝑧 ) ) → ¬ 𝑦 ∈ 𝑥 )
14 2 13 eximii ⊢ ∃ 𝑦 ¬ 𝑦 ∈ 𝑥
15 1 14 mpgbi ⊢ ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥