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Theorem uniex2 4515
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  |-  E. y 
y  =  U. x
Distinct variable group:    x, y

Proof of Theorem uniex2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 zfun 4513 . . . 4  |-  E. y A. z ( E. y
( z  e.  y  /\  y  e.  x
)  ->  z  e.  y )
2 eluni 3830 . . . . . . 7  |-  ( z  e.  U. x  <->  E. y
( z  e.  y  /\  y  e.  x
) )
32imbi1i 315 . . . . . 6  |-  ( ( z  e.  U. x  ->  z  e.  y )  <-> 
( E. y ( z  e.  y  /\  y  e.  x )  ->  z  e.  y ) )
43albii 1553 . . . . 5  |-  ( A. z ( z  e. 
U. x  ->  z  e.  y )  <->  A. z
( E. y ( z  e.  y  /\  y  e.  x )  ->  z  e.  y ) )
54exbii 1569 . . . 4  |-  ( E. y A. z ( z  e.  U. x  ->  z  e.  y )  <->  E. y A. z ( E. y ( z  e.  y  /\  y  e.  x )  ->  z  e.  y ) )
61, 5mpbir 200 . . 3  |-  E. y A. z ( z  e. 
U. x  ->  z  e.  y )
76bm1.3ii 4144 . 2  |-  E. y A. z ( z  e.  y  <->  z  e.  U. x )
8 dfcleq 2277 . . 3  |-  ( y  =  U. x  <->  A. z
( z  e.  y  <-> 
z  e.  U. x
) )
98exbii 1569 . 2  |-  ( E. y  y  =  U. x 
<->  E. y A. z
( z  e.  y  <-> 
z  e.  U. x
) )
107, 9mpbir 200 1  |-  E. y 
y  =  U. x
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   U.cuni 3827
This theorem is referenced by:  uniex  4516
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-uni 3828
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