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Theorem unexb 4177
 Description: Existence of union is equivalent to existence of its components. (Contributed by NM, 11-Jun-1998.)
Assertion
Ref Expression
unexb

Proof of Theorem unexb
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3090 . . . 4
21eleq1d 2106 . . 3
3 uneq2 3091 . . . 4
43eleq1d 2106 . . 3
5 vex 2560 . . . 4
6 vex 2560 . . . 4
75, 6unex 4176 . . 3
82, 4, 7vtocl2g 2617 . 2
9 ssun1 3106 . . . 4
10 ssexg 3896 . . . 4
119, 10mpan 400 . . 3
12 ssun2 3107 . . . 4
13 ssexg 3896 . . . 4
1412, 13mpan 400 . . 3
1511, 14jca 290 . 2
168, 15impbii 117 1
 Colors of variables: wff set class Syntax hints:   wa 97   wb 98   wceq 1243   wcel 1393  cvv 2557   cun 2915   wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581 This theorem is referenced by:  unexg  4178  sucexb  4223  frecabex  5984
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