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Theorem unipr 4262
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 1693 . . . 4
2 vex 3112 . . . . . . . 8
32elpr 4047 . . . . . . 7
43anbi2i 694 . . . . . 6
5 andi 867 . . . . . 6
64, 5bitri 249 . . . . 5
76exbii 1667 . . . 4
8 unipr.1 . . . . . . 7
98clel3 3238 . . . . . 6
10 exancom 1671 . . . . . 6
119, 10bitri 249 . . . . 5
12 unipr.2 . . . . . . 7
1312clel3 3238 . . . . . 6
14 exancom 1671 . . . . . 6
1513, 14bitri 249 . . . . 5
1611, 15orbi12i 521 . . . 4
171, 7, 163bitr4ri 278 . . 3
1817abbii 2591 . 2
19 df-un 3480 . 2
20 df-uni 4250 . 2
2118, 19, 203eqtr4ri 2497 1
Colors of variables: wff setvar class
Syntax hints:  \/wo 368  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442   cvv 3109  u.cun 3473  {cpr 4031  U.cuni 4249
This theorem is referenced by:  uniprg  4263  unisn  4264  uniintsn  4324  uniop  4755  unex  6598  rankxplim  8318  mrcun  15019  indistps  19512  indistps2  19513  leordtval2  19713  ex-uni  25147  fouriersw  32014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-sn 4030  df-pr 4032  df-uni 4250
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