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Theorem uniex2 4111
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  U.
Distinct variable group:   ,

Proof of Theorem uniex2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 zfun 4109 . . . 4
2 eluni 3546 . . . . . . 7  U.
32imbi1i 227 . . . . . 6  U.
43albii 1332 . . . . 5  U.
54exbii 1469 . . . 4  U.
61, 5mpbir 134 . . 3 
U.
76bm1.3ii 3841 . 2  U.
8 dfcleq 2007 . . 3  U.  U.
98exbii 1469 . 2  U.  U.
107, 9mpbir 134 1  U.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1221   wceq 1223  wex 1354   wcel 1366   U.cuni 3543
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-un 4108
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-uni 3544
This theorem is referenced by:  uniex  4112
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