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Theorem bj-nalset 32472
Description: Remove dependency on ax-13 2245 from nalset 4760. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nalset ¬ ∃𝑥𝑦 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-nalset
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 alexn 1768 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-sep 4746 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
3 elequ1 1994 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
4 elequ1 1994 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
5 elequ1 1994 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
6 elequ2 2001 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
75, 6bitrd 268 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
87notbid 308 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
94, 8anbi12d 746 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
103, 9bibi12d 335 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1110bj-spvv 32400 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
12 pclem6 970 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1311, 12syl 17 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
142, 13eximii 1761 . 2 𝑦 ¬ 𝑦𝑥
151, 14mpgbi 1722 1 ¬ ∃𝑥𝑦 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-sep 4746
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702
This theorem is referenced by: (None)
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