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Theorem unipr 3745
 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 1607 . . . 4
2 vex 2684 . . . . . . . 8
32elpr 3543 . . . . . . 7
43anbi2i 452 . . . . . 6
5 andi 807 . . . . . 6
64, 5bitri 183 . . . . 5
76exbii 1584 . . . 4
8 unipr.1 . . . . . . 7
98clel3 2815 . . . . . 6
10 exancom 1587 . . . . . 6
119, 10bitri 183 . . . . 5
12 unipr.2 . . . . . . 7
1312clel3 2815 . . . . . 6
14 exancom 1587 . . . . . 6
1513, 14bitri 183 . . . . 5
1611, 15orbi12i 753 . . . 4
171, 7, 163bitr4ri 212 . . 3
1817abbii 2253 . 2
19 df-un 3070 . 2
20 df-uni 3732 . 2
2118, 19, 203eqtr4ri 2169 1
 Colors of variables: wff set class Syntax hints:   wa 103   wo 697   wceq 1331  wex 1468   wcel 1480  cab 2123  cvv 2681   cun 3064  cpr 3523  cuni 3731 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529  df-uni 3732 This theorem is referenced by:  uniprg  3746  unisn  3747  uniop  4172  unex  4357  bj-unex  13106
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