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Theorem nalset 4128
Description: No set contains all sets. Theorem 41 of [Suppes] p. 30. (Contributed by NM, 23-Aug-1993.)
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 alexnim 1646 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 → ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-sep 4116 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
3 elequ1 2150 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
4 elequ1 2150 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
5 elequ1 2150 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
6 elequ2 2151 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
75, 6bitrd 188 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
87notbid 667 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
94, 8anbi12d 473 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
103, 9bibi12d 235 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1110spv 1858 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
12 pclem6 1374 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1311, 12syl 14 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
142, 13eximii 1600 . 2 𝑦 ¬ 𝑦𝑥
151, 14mpg 1449 1 ¬ ∃𝑥𝑦 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wal 1351  wex 1490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-5 1445  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-13 2148  ax-14 2149  ax-sep 4116
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359  df-nf 1459
This theorem is referenced by:  vnex  4129
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