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Theorem bj-nalset 10978
Description: nalset 3929 from bounded separation. (Contributed by BJ, 18-Nov-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 alexnim 1580 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 → ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-bdel 10904 . . . . 5 BOUNDED 𝑧𝑧
32ax-bdn 10900 . . . 4 BOUNDED ¬ 𝑧𝑧
43bdsep1 10968 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
5 elequ1 1642 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
6 elequ1 1642 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
7 elequ1 1642 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
8 elequ2 1643 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
97, 8bitrd 186 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
109notbid 625 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
116, 10anbi12d 457 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
125, 11bibi12d 233 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1312spv 1783 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
14 pclem6 1306 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1513, 14syl 14 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
164, 15eximii 1534 . 2 𝑦 ¬ 𝑦𝑥
171, 16mpg 1381 1 ¬ ∃𝑥𝑦 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 102  wb 103  wal 1283  wex 1422
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 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-bdn 10900  ax-bdel 10904  ax-bdsep 10967
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391
This theorem is referenced by:  bj-vprc  10979
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