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Theorem bj-unexg 13104
Description: unexg 4359 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 3218 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
2 eleq1 2200 . . 3 ((𝑥𝑦) = (𝐴𝑦) → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
31, 2syl 14 . 2 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
4 uneq2 3219 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
5 eleq1 2200 . . 3 ((𝐴𝑦) = (𝐴𝐵) → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
64, 5syl 14 . 2 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
7 vex 2684 . . 3 𝑥 ∈ V
8 vex 2684 . . 3 𝑦 ∈ V
97, 8bj-unex 13102 . 2 (𝑥𝑦) ∈ V
103, 6, 9vtocl2g 2745 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  Vcvv 2681  cun 3064
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-pr 4126  ax-un 4350  ax-bd0 12996  ax-bdor 12999  ax-bdex 13002  ax-bdeq 13003  ax-bdel 13004  ax-bdsb 13005  ax-bdsep 13067
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732  df-bdc 13024
This theorem is referenced by:  bj-sucexg  13105
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