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Theorem dfac5lem2 8517
Description: Lemma for dfac5 8521. (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 4260 . . 3  |-  U. A  =  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
32eleq2i 2545 . 2  |-  ( <.
w ,  g >.  e.  U. A  <->  <. w ,  g >.  e.  U. {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
4 eluniab 4262 . . 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 3021 . . . . 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 649 . . . . 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 250 . . . 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 1644 . . 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 3138 . . . 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 2823 . . . 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 251 . . 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 271 . 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 450 . . . . . . . . 9  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
14 ne0i 3796 . . . . . . . . . . 11  |-  ( <.
w ,  g >.  e.  u  ->  u  =/=  (/) )
1514pm4.71i 632 . . . . . . . . . 10  |-  ( <.
w ,  g >.  e.  u  <->  ( <. w ,  g >.  e.  u  /\  u  =/=  (/) ) )
1615anbi2i 694 . . . . . . . . 9  |-  ( ( u  =  ( { t }  X.  t
)  /\  <. w ,  g >.  e.  u
)  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
1713, 16bitr4i 252 . . . . . . . 8  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u ) )
1817exbii 1644 . . . . . . 7  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u ( u  =  ( { t }  X.  t )  /\  <. w ,  g
>.  e.  u ) )
19 snex 4694 . . . . . . . . 9  |-  { t }  e.  _V
20 vex 3121 . . . . . . . . 9  |-  t  e. 
_V
2119, 20xpex 6599 . . . . . . . 8  |-  ( { t }  X.  t
)  e.  _V
22 eleq2 2540 . . . . . . . 8  |-  ( u  =  ( { t }  X.  t )  ->  ( <. w ,  g >.  e.  u  <->  <.
w ,  g >.  e.  ( { t }  X.  t ) ) )
2321, 22ceqsexv 3155 . . . . . . 7  |-  ( E. u ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u )  <->  <. w ,  g >.  e.  ( { t }  X.  t ) )
2418, 23bitri 249 . . . . . 6  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  <. w ,  g
>.  e.  ( { t }  X.  t ) )
2524anbi2i 694 . . . . 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 5035 . . . . . . 7  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( w  e.  { t }  /\  g  e.  t )
)
27 elsn 4047 . . . . . . . . 9  |-  ( w  e.  { t }  <-> 
w  =  t )
28 equcom 1743 . . . . . . . . 9  |-  ( w  =  t  <->  t  =  w )
2927, 28bitri 249 . . . . . . . 8  |-  ( w  e.  { t }  <-> 
t  =  w )
3029anbi1i 695 . . . . . . 7  |-  ( ( w  e.  { t }  /\  g  e.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3126, 30bitri 249 . . . . . 6  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3231anbi2i 694 . . . . 5  |-  ( ( t  e.  h  /\  <.
w ,  g >.  e.  ( { t }  X.  t ) )  <-> 
( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) ) )
33 an12 795 . . . . 5  |-  ( ( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3425, 32, 333bitri 271 . . . 4  |-  ( ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3534exbii 1644 . . 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 3121 . . . 4  |-  w  e. 
_V
37 elequ1 1770 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
38 eleq2 2540 . . . . 5  |-  ( t  =  w  ->  (
g  e.  t  <->  g  e.  w ) )
3937, 38anbi12d 710 . . . 4  |-  ( t  =  w  ->  (
( t  e.  h  /\  g  e.  t
)  <->  ( w  e.  h  /\  g  e.  w ) ) )
4036, 39ceqsexv 3155 . . 3  |-  ( E. t ( t  =  w  /\  ( t  e.  h  /\  g  e.  t ) )  <->  ( w  e.  h  /\  g  e.  w ) )
4135, 40bitri 249 . 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 271 1  |-  ( <.
w ,  g >.  e.  U. A  <->  ( w  e.  h  /\  g  e.  w ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452    =/= wne 2662   E.wrex 2818   (/)c0 3790   {csn 4033   <.cop 4039   U.cuni 4251    X. cxp 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-opab 4512  df-xp 5011
This theorem is referenced by:  dfac5lem5  8520
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