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Theorem bj-unexg 13956
Description: unexg 4428 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 3274 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
2 eleq1 2233 . . 3 ((𝑥𝑦) = (𝐴𝑦) → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
31, 2syl 14 . 2 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
4 uneq2 3275 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
5 eleq1 2233 . . 3 ((𝐴𝑦) = (𝐴𝐵) → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
64, 5syl 14 . 2 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
7 vex 2733 . . 3 𝑥 ∈ V
8 vex 2733 . . 3 𝑦 ∈ V
97, 8bj-unex 13954 . 2 (𝑥𝑦) ∈ V
103, 6, 9vtocl2g 2794 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wcel 2141  Vcvv 2730  cun 3119
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-pr 4194  ax-un 4418  ax-bd0 13848  ax-bdor 13851  ax-bdex 13854  ax-bdeq 13855  ax-bdel 13856  ax-bdsb 13857  ax-bdsep 13919
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-uni 3797  df-bdc 13876
This theorem is referenced by:  bj-sucexg  13957
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