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Theorem bj-nalset 13308
 Description: nalset 4067 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 1628 . 2 (∀𝑥𝑦 ¬ 𝑦𝑥 → ¬ ∃𝑥𝑦 𝑦𝑥)
2 ax-bdel 13234 . . . . 5 BOUNDED 𝑧𝑧
32ax-bdn 13230 . . . 4 BOUNDED ¬ 𝑧𝑧
43bdsep1 13298 . . 3 𝑦𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧))
5 elequ1 1691 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝑦𝑦𝑦))
6 elequ1 1691 . . . . . . 7 (𝑧 = 𝑦 → (𝑧𝑥𝑦𝑥))
7 elequ1 1691 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑧))
8 elequ2 1692 . . . . . . . . 9 (𝑧 = 𝑦 → (𝑦𝑧𝑦𝑦))
97, 8bitrd 187 . . . . . . . 8 (𝑧 = 𝑦 → (𝑧𝑧𝑦𝑦))
109notbid 657 . . . . . . 7 (𝑧 = 𝑦 → (¬ 𝑧𝑧 ↔ ¬ 𝑦𝑦))
116, 10anbi12d 465 . . . . . 6 (𝑧 = 𝑦 → ((𝑧𝑥 ∧ ¬ 𝑧𝑧) ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
125, 11bibi12d 234 . . . . 5 (𝑧 = 𝑦 → ((𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) ↔ (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦))))
1312spv 1833 . . . 4 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → (𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)))
14 pclem6 1353 . . . 4 ((𝑦𝑦 ↔ (𝑦𝑥 ∧ ¬ 𝑦𝑦)) → ¬ 𝑦𝑥)
1513, 14syl 14 . . 3 (∀𝑧(𝑧𝑦 ↔ (𝑧𝑥 ∧ ¬ 𝑧𝑧)) → ¬ 𝑦𝑥)
164, 15eximii 1582 . 2 𝑦 ¬ 𝑦𝑥
171, 16mpg 1428 1 ¬ ∃𝑥𝑦 𝑦𝑥
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 103   ↔ wb 104  ∀wal 1330  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-bdn 13230  ax-bdel 13234  ax-bdsep 13297 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438 This theorem is referenced by:  bj-vprc  13309
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