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Theorem unipr 3741
 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
StepHypRef Expression
1 19.43 1604 . . . 4
2 vex 2730 . . . . . . . 8
32elpr 3562 . . . . . . 7
43anbi2i 678 . . . . . 6
5 andi 842 . . . . . 6
64, 5bitri 242 . . . . 5
76exbii 1580 . . . 4
8 unipr.1 . . . . . . 7
98clel3 2843 . . . . . 6
10 exancom 1584 . . . . . 6
119, 10bitri 242 . . . . 5
12 unipr.2 . . . . . . 7
1312clel3 2843 . . . . . 6
14 exancom 1584 . . . . . 6
1513, 14bitri 242 . . . . 5
1611, 15orbi12i 509 . . . 4
171, 7, 163bitr4ri 271 . . 3
1817abbii 2361 . 2
19 df-un 3083 . 2
20 df-uni 3728 . 2
2118, 19, 203eqtr4ri 2284 1
 Colors of variables: wff set class Syntax hints:   wo 359   wa 360  wex 1537   wceq 1619   wcel 1621  cab 2239  cvv 2727   cun 3076  cpr 3545  cuni 3727 This theorem is referenced by:  uniprg  3742  unisn  3743  uniintsn  3797  uniop  4162  unex  4409  rankxplim  7433  mrcun  13396  indistps  16580  indistps2  16581  leordtval2  16774  ex-uni  20626  toplat  24456 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-un 3083  df-sn 3550  df-pr 3551  df-uni 3728
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