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Theorem uniex2 4139
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 4137 . . . 4
2 eluni 3574 . . . . . . 7  U.
32imbi1i 227 . . . . . 6  U.
43albii 1356 . . . . 5  U.
54exbii 1493 . . . 4  U.
61, 5mpbir 134 . . 3 
U.
76bm1.3ii 3869 . 2  U.
8 dfcleq 2031 . . 3  U.  U.
98exbii 1493 . 2  U.  U.
107, 9mpbir 134 1  U.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98  wal 1240   wceq 1242  wex 1378   wcel 1390   U.cuni 3571
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-uni 3572
This theorem is referenced by:  uniex  4140
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