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Theorem bj-nalset 9888
Description: nalset 3884 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 1539 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 → ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-bdel 9814 . . . . 5 BOUNDED 𝑧𝑧
32ax-bdn 9810 . . . 4 BOUNDED ¬ 𝑧𝑧
43bdsep1 9878 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
5 elequ1 1600 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
6 elequ1 1600 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
7 elequ1 1600 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
8 elequ2 1601 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
97, 8bitrd 177 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
109notbid 592 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
116, 10anbi12d 442 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
125, 11bibi12d 224 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1312spv 1740 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
14 pclem6 1265 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1513, 14syl 14 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
164, 15eximii 1493 . 2 𝑦 ¬ 𝑦𝑥
171, 16mpg 1340 1 ¬ ∃𝑥𝑦 𝑦𝑥
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wb 98  wal 1241  wex 1381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-bdn 9810  ax-bdel 9814  ax-bdsep 9877
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350
This theorem is referenced by:  bj-vprc  9889
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