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Theorem unex 4142
 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 3585 . 2 {A, B} = (AB)
4 prexgOLD 3937 . . . 4 ((A V B V) → {A, B} V)
51, 2, 4mp2an 402 . . 3 {A, B} V
65uniex 4140 . 2 {A, B} V
73, 6eqeltrri 2108 1 (AB) V
 Colors of variables: wff set class Syntax hints:   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909  {cpr 3368  ∪ cuni 3571 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-bndl 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-sn 3373  df-pr 3374  df-uni 3572 This theorem is referenced by:  unexb  4143  rdg0  5914  unen  6229  nn0ex  7963  xrex  8526
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