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Theorem bj-uniex 14909
Description: uniex 4449 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-uniex.1 𝐴 ∈ V
Assertion
Ref Expression
bj-uniex 𝐴 ∈ V

Proof of Theorem bj-uniex
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-uniex.1 . 2 𝐴 ∈ V
2 unieq 3830 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
32eleq1d 2256 . 2 (𝑥 = 𝐴 → ( 𝑥 ∈ V ↔ 𝐴 ∈ V))
4 bj-uniex2 14908 . . 3 𝑦 𝑦 = 𝑥
54issetri 2758 . 2 𝑥 ∈ V
61, 3, 5vtocl 2803 1 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1363  wcel 2158  Vcvv 2749   cuni 3821
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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-un 4445  ax-bd0 14805  ax-bdex 14811  ax-bdel 14813  ax-bdsb 14814  ax-bdsep 14876
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-uni 3822  df-bdc 14833
This theorem is referenced by:  bj-uniexg  14910  bj-unex  14911
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