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Theorem unexg 4443
Description: A union of two sets is a set. Corollary 5.8 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Sep-2006.)
Assertion
Ref Expression
unexg ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)

Proof of Theorem unexg
StepHypRef Expression
1 elex 2748 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 2748 . 2 (𝐵𝑊𝐵 ∈ V)
3 unexb 4442 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
43biimpi 120 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
51, 2, 4syl2an 289 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2148  Vcvv 2737  cun 3127
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810
This theorem is referenced by:  tpexg  4444  eldifpw  4477  ifelpwung  4481  xpexg  4740  tposexg  6258  tfrlemisucaccv  6325  tfrlemibxssdm  6327  tfrlemibfn  6328  tfr1onlemsucaccv  6341  tfr1onlembxssdm  6343  tfr1onlembfn  6344  tfrcllemsucaccv  6354  tfrcllembxssdm  6356  tfrcllembfn  6357  rdgtfr  6374  rdgruledefgg  6375  rdgivallem  6381  djuex  7041  zfz1isolem1  10819  ennnfonelemp1  12406  setsvalg  12491  setsex  12493  setsslid  12512  strleund  12561  prdsex  12717
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