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Theorem unexb 4701
 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 3486 . . . 4
21eleq1d 2501 . . 3
3 uneq2 3487 . . . 4
43eleq1d 2501 . . 3
5 vex 2951 . . . 4
6 vex 2951 . . . 4
75, 6unex 4699 . . 3
82, 4, 7vtocl2g 3007 . 2
9 ssun1 3502 . . . 4
10 ssexg 4341 . . . 4
119, 10mpan 652 . . 3
12 ssun2 3503 . . . 4
13 ssexg 4341 . . . 4
1412, 13mpan 652 . . 3
1511, 14jca 519 . 2
168, 15impbii 181 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359   wceq 1652   wcel 1725  cvv 2948   cun 3310   wss 3312 This theorem is referenced by:  unexg  4702  sucexb  4781  fodomr  7250  cdaval  8042 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812  df-pr 3813  df-uni 4008
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