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Theorem uniex2 4199
 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 4197 . . . 4
2 eluni 3612 . . . . . . 7
32imbi1i 236 . . . . . 6
43albii 1400 . . . . 5
54exbii 1537 . . . 4
61, 5mpbir 144 . . 3
76bm1.3ii 3907 . 2
8 dfcleq 2076 . . 3
98exbii 1537 . 2
107, 9mpbir 144 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103  wal 1283   wceq 1285  wex 1422   wcel 1434  cuni 3609 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-un 4196 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-uni 3610 This theorem is referenced by:  uniex  4200
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