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Theorem nalset 5193
 Description: No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.) Remove use of ax-12 2178 and ax-13 2391. (Revised by BJ, 31-May-2019.)
Assertion
Ref Expression
nalset ¬ ∃𝑥𝑦 𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem nalset
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 alexn 1846 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 ↔ ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-sep 5179 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
3 elequ1 2121 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
4 elequ1 2121 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
5 elequ1 2121 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
6 elequ2 2129 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
75, 6bitrd 282 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
87notbid 321 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
94, 8anbi12d 633 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
103, 9bibi12d 349 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1110spvv 2003 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
12 pclem6 1023 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1311, 12syl 17 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
142, 13eximii 1838 . 2 𝑦 ¬ 𝑦𝑥
151, 14mpgbi 1800 1 ¬ ∃𝑥𝑦 𝑦𝑥
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-sep 5179 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  vnex  5194  kmlem2  9566  iota0ndef  43570  aiota0ndef  43591
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