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Theorem unipr 3622
 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 1535 . . . 4
2 vex 2577 . . . . . . . 8
32elpr 3424 . . . . . . 7
43anbi2i 438 . . . . . 6
5 andi 742 . . . . . 6
64, 5bitri 177 . . . . 5
76exbii 1512 . . . 4
8 unipr.1 . . . . . . 7
98clel3 2702 . . . . . 6
10 exancom 1515 . . . . . 6
119, 10bitri 177 . . . . 5
12 unipr.2 . . . . . . 7
1312clel3 2702 . . . . . 6
14 exancom 1515 . . . . . 6
1513, 14bitri 177 . . . . 5
1611, 15orbi12i 691 . . . 4
171, 7, 163bitr4ri 206 . . 3
1817abbii 2169 . 2
19 df-un 2950 . 2
20 df-uni 3609 . 2
2118, 19, 203eqtr4ri 2087 1
 Colors of variables: wff set class Syntax hints:   wa 101   wo 639   wceq 1259  wex 1397   wcel 1409  cab 2042  cvv 2574   cun 2943  cpr 3404  cuni 3608 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-uni 3609 This theorem is referenced by:  uniprg  3623  unisn  3624  uniop  4020  unex  4204  bj-unex  10426
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