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Theorem dfac5lem2 7767
Description: Lemma for dfac5 7771. (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 3853 . . 3  |-  U. A  =  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
32eleq2i 2360 . 2  |-  ( <.
w ,  g >.  e.  U. A  <->  <. w ,  g >.  e.  U. {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
4 eluniab 3855 . . 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 2707 . . . . 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 630 . . . . 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 241 . . . 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 1572 . . 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 2820 . . . 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 2562 . . . 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 242 . . 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 262 . 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 437 . . . . . . . . 9  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
14 ne0i 3474 . . . . . . . . . . 11  |-  ( <.
w ,  g >.  e.  u  ->  u  =/=  (/) )
1514pm4.71i 613 . . . . . . . . . 10  |-  ( <.
w ,  g >.  e.  u  <->  ( <. w ,  g >.  e.  u  /\  u  =/=  (/) ) )
1615anbi2i 675 . . . . . . . . 9  |-  ( ( u  =  ( { t }  X.  t
)  /\  <. w ,  g >.  e.  u
)  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
1713, 16bitr4i 243 . . . . . . . 8  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u ) )
1817exbii 1572 . . . . . . 7  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u ( u  =  ( { t }  X.  t )  /\  <. w ,  g
>.  e.  u ) )
19 snex 4232 . . . . . . . . 9  |-  { t }  e.  _V
20 vex 2804 . . . . . . . . 9  |-  t  e. 
_V
2119, 20xpex 4817 . . . . . . . 8  |-  ( { t }  X.  t
)  e.  _V
22 eleq2 2357 . . . . . . . 8  |-  ( u  =  ( { t }  X.  t )  ->  ( <. w ,  g >.  e.  u  <->  <.
w ,  g >.  e.  ( { t }  X.  t ) ) )
2321, 22ceqsexv 2836 . . . . . . 7  |-  ( E. u ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u )  <->  <. w ,  g >.  e.  ( { t }  X.  t ) )
2418, 23bitri 240 . . . . . 6  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  <. w ,  g
>.  e.  ( { t }  X.  t ) )
2524anbi2i 675 . . . . 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 4735 . . . . . . 7  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( w  e.  { t }  /\  g  e.  t )
)
27 elsn 3668 . . . . . . . . 9  |-  ( w  e.  { t }  <-> 
w  =  t )
28 equcom 1665 . . . . . . . . 9  |-  ( w  =  t  <->  t  =  w )
2927, 28bitri 240 . . . . . . . 8  |-  ( w  e.  { t }  <-> 
t  =  w )
3029anbi1i 676 . . . . . . 7  |-  ( ( w  e.  { t }  /\  g  e.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3126, 30bitri 240 . . . . . 6  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3231anbi2i 675 . . . . 5  |-  ( ( t  e.  h  /\  <.
w ,  g >.  e.  ( { t }  X.  t ) )  <-> 
( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) ) )
33 an12 772 . . . . 5  |-  ( ( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3425, 32, 333bitri 262 . . . 4  |-  ( ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3534exbii 1572 . . 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 2804 . . . 4  |-  w  e. 
_V
37 elequ1 1699 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
38 eleq2 2357 . . . . 5  |-  ( t  =  w  ->  (
g  e.  t  <->  g  e.  w ) )
3937, 38anbi12d 691 . . . 4  |-  ( t  =  w  ->  (
( t  e.  h  /\  g  e.  t
)  <->  ( w  e.  h  /\  g  e.  w ) ) )
4036, 39ceqsexv 2836 . . 3  |-  ( E. t ( t  =  w  /\  ( t  e.  h  /\  g  e.  t ) )  <->  ( w  e.  h  /\  g  e.  w ) )
4135, 40bitri 240 . 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 262 1  |-  ( <.
w ,  g >.  e.  U. A  <->  ( w  e.  h  /\  g  e.  w ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   (/)c0 3468   {csn 3653   <.cop 3656   U.cuni 3843    X. cxp 4703
This theorem is referenced by:  dfac5lem5  7770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-opab 4094  df-xp 4711
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