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Theorem uniex2 4139
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 4137 . . . 4 yz(y(z y y x) → z y)
2 eluni 3574 . . . . . . 7 (z xy(z y y x))
32imbi1i 227 . . . . . 6 ((z xz y) ↔ (y(z y y x) → z y))
43albii 1356 . . . . 5 (z(z xz y) ↔ z(y(z y y x) → z y))
54exbii 1493 . . . 4 (yz(z xz y) ↔ yz(y(z y y x) → z y))
61, 5mpbir 134 . . 3 yz(z xz y)
76bm1.3ii 3869 . 2 yz(z yz x)
8 dfcleq 2031 . . 3 (y = xz(z yz x))
98exbii 1493 . 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 1240   = wceq 1242  wex 1378   wcel 1390   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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