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Theorem uniex2 4696
 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 4694 . . . 4
2 eluni 4010 . . . . . . 7
32imbi1i 316 . . . . . 6
43albii 1575 . . . . 5
54exbii 1592 . . . 4
61, 5mpbir 201 . . 3
76bm1.3ii 4325 . 2
8 dfcleq 2429 . . 3
98exbii 1592 . 2
107, 9mpbir 201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  cuni 4007 This theorem is referenced by:  uniex  4697 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-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-v 2950  df-uni 4008
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