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Theorem unexg 4444
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 2749 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 2749 . 2 (𝐵𝑊𝐵 ∈ V)
3 unexb 4443 . . 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 2738  cun 3128
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 4122  ax-pr 4210  ax-un 4434
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 2740  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-uni 3811
This theorem is referenced by:  tpexg  4445  eldifpw  4478  ifelpwung  4482  xpexg  4741  tposexg  6259  tfrlemisucaccv  6326  tfrlemibxssdm  6328  tfrlemibfn  6329  tfr1onlemsucaccv  6342  tfr1onlembxssdm  6344  tfr1onlembfn  6345  tfrcllemsucaccv  6355  tfrcllembxssdm  6357  tfrcllembfn  6358  rdgtfr  6375  rdgruledefgg  6376  rdgivallem  6382  djuex  7042  zfz1isolem1  10820  ennnfonelemp1  12407  setsvalg  12492  setsex  12494  setsslid  12513  strleund  12562  prdsex  12718
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