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Theorem unexb 6605
 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 3613 . . . 4
21eleq1d 2491 . . 3
3 uneq2 3614 . . . 4
43eleq1d 2491 . . 3
5 vex 3083 . . . 4
6 vex 3083 . . . 4
75, 6unex 6603 . . 3
82, 4, 7vtocl2g 3143 . 2
9 ssun1 3629 . . . 4
10 ssexg 4570 . . . 4
119, 10mpan 674 . . 3
12 ssun2 3630 . . . 4
13 ssexg 4570 . . . 4
1412, 13mpan 674 . . 3
1511, 14jca 534 . 2
168, 15impbii 190 1
 Colors of variables: wff setvar class Syntax hints:   wb 187   wa 370   wceq 1437   wcel 1872  cvv 3080   cun 3434   wss 3436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-rex 2777  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-sn 3999  df-pr 4001  df-uni 4220 This theorem is referenced by:  unexg  6606  sucexb  6650  fodomr  7732  fsuppun  7911  fsuppunbi  7913  cdaval  8607  bj-tagex  31549
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