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Theorem bj-uniex 13222
Description: uniex 4359 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 3745 . . 3 (𝑥 = 𝐴 𝑥 = 𝐴)
32eleq1d 2208 . 2 (𝑥 = 𝐴 → ( 𝑥 ∈ V ↔ 𝐴 ∈ V))
4 bj-uniex2 13221 . . 3 𝑦 𝑦 = 𝑥
54issetri 2695 . 2 𝑥 ∈ V
61, 3, 5vtocl 2740 1 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  Vcvv 2686   cuni 3736
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 2121  ax-un 4355  ax-bd0 13118  ax-bdex 13124  ax-bdel 13126  ax-bdsb 13127  ax-bdsep 13189
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-uni 3737  df-bdc 13146
This theorem is referenced by:  bj-uniexg  13223  bj-unex  13224
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