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Theorem unipr 4031
 Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 23-Aug-1993.)
Hypotheses
Ref Expression
unipr.1
unipr.2
Assertion
Ref Expression
unipr

Proof of Theorem unipr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.43 1616 . . . 4
2 vex 2961 . . . . . . . 8
32elpr 3834 . . . . . . 7
43anbi2i 677 . . . . . 6
5 andi 839 . . . . . 6
64, 5bitri 242 . . . . 5
76exbii 1593 . . . 4
8 unipr.1 . . . . . . 7
98clel3 3076 . . . . . 6
10 exancom 1597 . . . . . 6
119, 10bitri 242 . . . . 5
12 unipr.2 . . . . . . 7
1312clel3 3076 . . . . . 6
14 exancom 1597 . . . . . 6
1513, 14bitri 242 . . . . 5
1611, 15orbi12i 509 . . . 4
171, 7, 163bitr4ri 271 . . 3
1817abbii 2550 . 2
19 df-un 3327 . 2
20 df-uni 4018 . 2
2118, 19, 203eqtr4ri 2469 1
 Colors of variables: wff set class Syntax hints:   wo 359   wa 360  wex 1551   wceq 1653   wcel 1726  cab 2424  cvv 2958   cun 3320  cpr 3817  cuni 4017 This theorem is referenced by:  uniprg  4032  unisn  4033  uniintsn  4089  uniop  4461  unex  4709  rankxplim  7805  mrcun  13849  indistps  17077  indistps2  17078  leordtval2  17278  ex-uni  21736 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-pr 3823  df-uni 4018
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