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Theorem uniex2 4121
Description: The Axiom of Union using the standard abbreviation for union. Given any set x, its union y exists. (Contributed by NM, 4-Jun-2006.)
Assertion
Ref Expression
uniex2 y y = x
Distinct variable group:   x,y

Proof of Theorem uniex2
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 zfun 4119 . . . 4 yz(y(z y y x) → z y)
2 eluni 3556 . . . . . . 7 (z xy(z y y x))
32imbi1i 227 . . . . . 6 ((z xz y) ↔ (y(z y y x) → z y))
43albii 1339 . . . . 5 (z(z xz y) ↔ z(y(z y y x) → z y))
54exbii 1480 . . . 4 (yz(z xz y) ↔ yz(y(z y y x) → z y))
61, 5mpbir 134 . . 3 yz(z xz y)
76bm1.3ii 3851 . 2 yz(z yz x)
8 dfcleq 2017 . . 3 (y = xz(z yz x))
98exbii 1480 . 2 (y y = xyz(z yz x))
107, 9mpbir 134 1 y y = x
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1226   = wceq 1228  wex 1363   wcel 1375   cuni 3553
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-uni 3554
This theorem is referenced by:  uniex  4122
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