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Theorem unexg 4144
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 ((A 𝑉 B 𝑊) → (AB) V)

Proof of Theorem unexg
StepHypRef Expression
1 elex 2560 . 2 (A 𝑉A V)
2 elex 2560 . 2 (B 𝑊B V)
3 unexb 4143 . . 3 ((A V B V) ↔ (AB) V)
43biimpi 113 . 2 ((A V B V) → (AB) V)
51, 2, 4syl2an 273 1 ((A 𝑉 B 𝑊) → (AB) V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  cun 2909
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572
This theorem is referenced by:  tpexg  4145  eldifpw  4174  xpexg  4395  tposexg  5814  tfrlemisucaccv  5880  tfrlemibxssdm  5882  tfrlemibfn  5883  rdgtfr  5901  rdgruledefgg  5902  rdgivallem  5908
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