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Theorem bj-nalset 32919
Description: Remove dependency on ax-13 2282 from nalset 4828. (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 1811 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-sep 4814 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
3 elequ1 2037 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
4 elequ1 2037 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
5 elequ1 2037 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
6 elequ2 2044 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
75, 6bitrd 268 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
87notbid 307 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
94, 8anbi12d 747 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
103, 9bibi12d 334 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1110bj-spvv 32848 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
12 pclem6 991 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1311, 12syl 17 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
142, 13eximii 1804 . 2 𝑦 ¬ 𝑦𝑥
151, 14mpgbi 1765 1 ¬ ∃𝑥𝑦 𝑦𝑥
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 383  wal 1521  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-sep 4814
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by: (None)
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