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Theorem bj-uniex 15066
Description: uniex 4452 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 3833 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
32eleq1d 2258 . 2 (𝑥 = 𝐴 → ( 𝑥 ∈ V ↔ 𝐴 ∈ V))
4 bj-uniex2 15065 . . 3 𝑦 𝑦 = 𝑥
54issetri 2761 . 2 𝑥 ∈ V
61, 3, 5vtocl 2806 1 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wcel 2160  Vcvv 2752   cuni 3824
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-un 4448  ax-bd0 14962  ax-bdex 14968  ax-bdel 14970  ax-bdsb 14971  ax-bdsep 15033
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-uni 3825  df-bdc 14990
This theorem is referenced by:  bj-uniexg  15067  bj-unex  15068
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