Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-unexg GIF version

Theorem bj-unexg 14555
Description: unexg 4443 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 3282 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
2 eleq1 2240 . . 3 ((𝑥𝑦) = (𝐴𝑦) → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
31, 2syl 14 . 2 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
4 uneq2 3283 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
5 eleq1 2240 . . 3 ((𝐴𝑦) = (𝐴𝐵) → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
64, 5syl 14 . 2 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
7 vex 2740 . . 3 𝑥 ∈ V
8 vex 2740 . . 3 𝑦 ∈ V
97, 8bj-unex 14553 . 2 (𝑥𝑦) ∈ V
103, 6, 9vtocl2g 2801 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  Vcvv 2737  cun 3127
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-pr 4209  ax-un 4433  ax-bd0 14447  ax-bdor 14450  ax-bdex 14453  ax-bdeq 14454  ax-bdel 14455  ax-bdsb 14456  ax-bdsep 14518
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-uni 3810  df-bdc 14475
This theorem is referenced by:  bj-sucexg  14556
  Copyright terms: Public domain W3C validator