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Theorem bj-unexg 11695
Description: unexg 4266 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-unexg ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem bj-unexg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3147 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
2 eleq1 2150 . . 3 ((𝑥𝑦) = (𝐴𝑦) → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
31, 2syl 14 . 2 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
4 uneq2 3148 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
5 eleq1 2150 . . 3 ((𝐴𝑦) = (𝐴𝐵) → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
64, 5syl 14 . 2 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
7 vex 2622 . . 3 𝑥 ∈ V
8 vex 2622 . . 3 𝑦 ∈ V
97, 8bj-unex 11693 . 2 (𝑥𝑦) ∈ V
103, 6, 9vtocl2g 2683 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619  cun 2997
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-pr 4034  ax-un 4258  ax-bd0 11587  ax-bdor 11590  ax-bdex 11593  ax-bdeq 11594  ax-bdel 11595  ax-bdsb 11596  ax-bdsep 11658
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-sn 3450  df-pr 3451  df-uni 3652  df-bdc 11615
This theorem is referenced by:  bj-sucexg  11696
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