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Theorem unex 4418
Description: The union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 1-Jul-1994.)
Hypotheses
Ref Expression
unex.1 𝐴 ∈ V
unex.2 𝐵 ∈ V
Assertion
Ref Expression
unex (𝐴𝐵) ∈ V

Proof of Theorem unex
StepHypRef Expression
1 unex.1 . . 3 𝐴 ∈ V
2 unex.2 . . 3 𝐵 ∈ V
31, 2unipr 3802 . 2 {𝐴, 𝐵} = (𝐴𝐵)
4 prexg 4188 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V)
51, 2, 4mp2an 423 . . 3 {𝐴, 𝐵} ∈ V
65uniex 4414 . 2 {𝐴, 𝐵} ∈ V
73, 6eqeltrri 2239 1 (𝐴𝐵) ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2136  Vcvv 2725  cun 3113  {cpr 3576   cuni 3788
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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pr 4186  ax-un 4410
This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449  df-v 2727  df-un 3119  df-sn 3581  df-pr 3582  df-uni 3789
This theorem is referenced by:  unexb  4419  rdg0  6351  unen  6778  findcard2  6851  findcard2s  6852  ac6sfi  6860  sbthlemi10  6927  finomni  7100  exmidfodomrlemim  7153  nn0ex  9116  xrex  9788  exmidunben  12355  strleun  12479
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