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Theorem nalset 3944
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 1582 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 → ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-sep 3932 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
3 elequ1 1644 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
4 elequ1 1644 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
5 elequ1 1644 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
6 elequ2 1645 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
75, 6bitrd 186 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
87notbid 625 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
94, 8anbi12d 457 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
103, 9bibi12d 233 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1110spv 1785 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
12 pclem6 1308 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1311, 12syl 14 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
142, 13eximii 1536 . 2 𝑦 ¬ 𝑦𝑥
151, 14mpg 1383 1 ¬ ∃𝑥𝑦 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wal 1285  wex 1424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-sep 3932
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393
This theorem is referenced by:  vnex  3945
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