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Theorem bj-unex 10039
 Description: unex 4176 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-unex.1 𝐴 ∈ V
bj-unex.2 𝐵 ∈ V
Assertion
Ref Expression
bj-unex (𝐴𝐵) ∈ V

Proof of Theorem bj-unex
StepHypRef Expression
1 bj-unex.1 . . 3 𝐴 ∈ V
2 bj-unex.2 . . 3 𝐵 ∈ V
31, 2unipr 3594 . 2 {𝐴, 𝐵} = (𝐴𝐵)
4 bj-prexg 10031 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
51, 2, 4mp2an 402 . . 3 {𝐴, 𝐵} ∈ V
65bj-uniex 10037 . 2 {𝐴, 𝐵} ∈ V
73, 6eqeltrri 2111 1 (𝐴𝐵) ∈ V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915  {cpr 3376  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-bdc 9961 This theorem is referenced by:  bdunexb  10040  bj-unexg  10041
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