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Theorem dfac5lem2 8553
Description: Lemma for dfac5 8557. (Contributed by NM, 12-Apr-2004.)
Hypothesis
Ref Expression
dfac5lem.1  |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
Assertion
Ref Expression
dfac5lem2  |-  ( <.
w ,  g >.  e.  U. A  <->  ( w  e.  h  /\  g  e.  w ) )
Distinct variable groups:    w, u, t, h, g    w, A, g
Allowed substitution hints:    A( u, t, h)

Proof of Theorem dfac5lem2
StepHypRef Expression
1 dfac5lem.1 . . . 4  |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
21unieqi 4231 . . 3  |-  U. A  =  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
32eleq2i 2507 . 2  |-  ( <.
w ,  g >.  e.  U. A  <->  <. w ,  g >.  e.  U. {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
4 eluniab 4233 . . 3  |-  ( <.
w ,  g >.  e.  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }  <->  E. u
( <. w ,  g
>.  e.  u  /\  (
u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) ) )
5 r19.42v 2990 . . . . 5  |-  ( E. t  e.  h  ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) )
6 anass 653 . . . . 5  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  E. t  e.  h  u  =  ( { t }  X.  t ) )  <->  ( <. w ,  g >.  e.  u  /\  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) ) )
75, 6bitr2i 253 . . . 4  |-  ( (
<. w ,  g >.  e.  u  /\  (
u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) )  <->  E. t  e.  h  ( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )
87exbii 1714 . . 3  |-  ( E. u ( <. w ,  g >.  e.  u  /\  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) )  <->  E. u E. t  e.  h  ( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( {
t }  X.  t
) ) )
9 rexcom4 3107 . . . 4  |-  ( E. t  e.  h  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u E. t  e.  h  ( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )
10 df-rex 2788 . . . 4  |-  ( E. t  e.  h  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. t ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) ) )
119, 10bitr3i 254 . . 3  |-  ( E. u E. t  e.  h  ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. t ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) ) )
124, 8, 113bitri 274 . 2  |-  ( <.
w ,  g >.  e.  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }  <->  E. t
( t  e.  h  /\  E. u ( (
<. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) ) )
13 ancom 451 . . . . . . . . 9  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
14 ne0i 3773 . . . . . . . . . . 11  |-  ( <.
w ,  g >.  e.  u  ->  u  =/=  (/) )
1514pm4.71i 636 . . . . . . . . . 10  |-  ( <.
w ,  g >.  e.  u  <->  ( <. w ,  g >.  e.  u  /\  u  =/=  (/) ) )
1615anbi2i 698 . . . . . . . . 9  |-  ( ( u  =  ( { t }  X.  t
)  /\  <. w ,  g >.  e.  u
)  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
1713, 16bitr4i 255 . . . . . . . 8  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u ) )
1817exbii 1714 . . . . . . 7  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u ( u  =  ( { t }  X.  t )  /\  <. w ,  g
>.  e.  u ) )
19 snex 4663 . . . . . . . . 9  |-  { t }  e.  _V
20 vex 3090 . . . . . . . . 9  |-  t  e. 
_V
2119, 20xpex 6609 . . . . . . . 8  |-  ( { t }  X.  t
)  e.  _V
22 eleq2 2502 . . . . . . . 8  |-  ( u  =  ( { t }  X.  t )  ->  ( <. w ,  g >.  e.  u  <->  <.
w ,  g >.  e.  ( { t }  X.  t ) ) )
2321, 22ceqsexv 3124 . . . . . . 7  |-  ( E. u ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u )  <->  <. w ,  g >.  e.  ( { t }  X.  t ) )
2418, 23bitri 252 . . . . . 6  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  <. w ,  g
>.  e.  ( { t }  X.  t ) )
2524anbi2i 698 . . . . 5  |-  ( ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )  <->  ( t  e.  h  /\  <. w ,  g >.  e.  ( { t }  X.  t ) ) )
26 opelxp 4884 . . . . . . 7  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( w  e.  { t }  /\  g  e.  t )
)
27 elsn 4016 . . . . . . . . 9  |-  ( w  e.  { t }  <-> 
w  =  t )
28 equcom 1846 . . . . . . . . 9  |-  ( w  =  t  <->  t  =  w )
2927, 28bitri 252 . . . . . . . 8  |-  ( w  e.  { t }  <-> 
t  =  w )
3029anbi1i 699 . . . . . . 7  |-  ( ( w  e.  { t }  /\  g  e.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3126, 30bitri 252 . . . . . 6  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3231anbi2i 698 . . . . 5  |-  ( ( t  e.  h  /\  <.
w ,  g >.  e.  ( { t }  X.  t ) )  <-> 
( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) ) )
33 an12 804 . . . . 5  |-  ( ( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3425, 32, 333bitri 274 . . . 4  |-  ( ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3534exbii 1714 . . 3  |-  ( E. t ( t  e.  h  /\  E. u
( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( {
t }  X.  t
) ) )  <->  E. t
( t  =  w  /\  ( t  e.  h  /\  g  e.  t ) ) )
36 vex 3090 . . . 4  |-  w  e. 
_V
37 elequ1 1873 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
38 eleq2 2502 . . . . 5  |-  ( t  =  w  ->  (
g  e.  t  <->  g  e.  w ) )
3937, 38anbi12d 715 . . . 4  |-  ( t  =  w  ->  (
( t  e.  h  /\  g  e.  t
)  <->  ( w  e.  h  /\  g  e.  w ) ) )
4036, 39ceqsexv 3124 . . 3  |-  ( E. t ( t  =  w  /\  ( t  e.  h  /\  g  e.  t ) )  <->  ( w  e.  h  /\  g  e.  w ) )
4135, 40bitri 252 . 2  |-  ( E. t ( t  e.  h  /\  E. u
( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( {
t }  X.  t
) ) )  <->  ( w  e.  h  /\  g  e.  w ) )
423, 12, 413bitri 274 1  |-  ( <.
w ,  g >.  e.  U. A  <->  ( w  e.  h  /\  g  e.  w ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414    =/= wne 2625   E.wrex 2783   (/)c0 3767   {csn 4002   <.cop 4008   U.cuni 4222    X. cxp 4852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-opab 4485  df-xp 4860  df-rel 4861
This theorem is referenced by:  dfac5lem5  8556
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