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Theorem uniex2 4119
 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 4117 . . . 4
2 eluni 3553 . . . . . . 7
32imbi1i 227 . . . . . 6
43albii 1335 . . . . 5
54exbii 1474 . . . 4
61, 5mpbir 134 . . 3
76bm1.3ii 3848 . 2
8 dfcleq 2012 . . 3
98exbii 1474 . 2
107, 9mpbir 134 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98  wal 1224   wceq 1226  wex 1358   wcel 1370  cuni 3550 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-un 4116 This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-uni 3551 This theorem is referenced by:  uniex  4120
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