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Theorem uniex2 6595
 Description: The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
uniex2
Distinct variable group:   ,

Proof of Theorem uniex2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 zfun 6593 . . . 4
2 eluni 4252 . . . . . . 7
32imbi1i 325 . . . . . 6
43albii 1640 . . . . 5
54exbii 1667 . . . 4
61, 5mpbir 209 . . 3
76bm1.3ii 4576 . 2
8 dfcleq 2450 . . 3
98exbii 1667 . 2
107, 9mpbir 209 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  E.wex 1612  e.wcel 1818  U.cuni 4249 This theorem is referenced by:  uniex  6596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-uni 4250
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