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Theorem unexg 4334
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 2671 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 2671 . 2 (𝐵𝑊𝐵 ∈ V)
3 unexb 4333 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
43biimpi 119 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
51, 2, 4syl2an 287 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1465  Vcvv 2660  cun 3039
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-uni 3707
This theorem is referenced by:  tpexg  4335  eldifpw  4368  xpexg  4623  tposexg  6123  tfrlemisucaccv  6190  tfrlemibxssdm  6192  tfrlemibfn  6193  tfr1onlemsucaccv  6206  tfr1onlembxssdm  6208  tfr1onlembfn  6209  tfrcllemsucaccv  6219  tfrcllembxssdm  6221  tfrcllembfn  6222  rdgtfr  6239  rdgruledefgg  6240  rdgivallem  6246  djuex  6896  zfz1isolem1  10551  ennnfonelemp1  11846  setsvalg  11916  setsex  11918  setsslid  11936  strleund  11974
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