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Theorem fineqvlem 7259
Description: Lemma for fineqv 7260. (Contributed by Mario Carneiro, 20-Jan-2013.) (Proof shortened by Stefan O'Rear, 3-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Assertion
Ref Expression
fineqvlem  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  om  ~<_  ~P ~P A )

Proof of Theorem fineqvlem
Dummy variables  b 
c  d  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwexg 4324 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 452 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ~P A  e. 
_V )
3 pwexg 4324 . . 3  |-  ( ~P A  e.  _V  ->  ~P ~P A  e.  _V )
42, 3syl 16 . 2  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ~P ~P A  e.  _V )
5 ssrab2 3371 . . . . 5  |-  { d  e.  ~P A  | 
d  ~~  b }  C_ 
~P A
6 elpw2g 4304 . . . . . 6  |-  ( ~P A  e.  _V  ->  ( { d  e.  ~P A  |  d  ~~  b }  e.  ~P ~P A  <->  { d  e.  ~P A  |  d  ~~  b }  C_  ~P A
) )
72, 6syl 16 . . . . 5  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( { d  e.  ~P A  | 
d  ~~  b }  e.  ~P ~P A  <->  { d  e.  ~P A  |  d 
~~  b }  C_  ~P A ) )
85, 7mpbiri 225 . . . 4  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  { d  e. 
~P A  |  d 
~~  b }  e.  ~P ~P A )
98a1d 23 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( b  e. 
om  ->  { d  e. 
~P A  |  d 
~~  b }  e.  ~P ~P A ) )
10 isinf 7258 . . . . . . . . 9  |-  ( -.  A  e.  Fin  ->  A. b  e.  om  E. e ( e  C_  A  /\  e  ~~  b
) )
1110r19.21bi 2747 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  b  e.  om )  ->  E. e ( e 
C_  A  /\  e  ~~  b ) )
1211ad2ant2lr 729 . . . . . . 7  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  E. e
( e  C_  A  /\  e  ~~  b ) )
13 vex 2902 . . . . . . . . . . . 12  |-  e  e. 
_V
1413elpw 3748 . . . . . . . . . . 11  |-  ( e  e.  ~P A  <->  e  C_  A )
1514biimpri 198 . . . . . . . . . 10  |-  ( e 
C_  A  ->  e  e.  ~P A )
1615anim1i 552 . . . . . . . . 9  |-  ( ( e  C_  A  /\  e  ~~  b )  -> 
( e  e.  ~P A  /\  e  ~~  b
) )
17 breq1 4156 . . . . . . . . . 10  |-  ( d  =  e  ->  (
d  ~~  b  <->  e  ~~  b ) )
1817elrab 3035 . . . . . . . . 9  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  <->  ( e  e.  ~P A  /\  e  ~~  b ) )
1916, 18sylibr 204 . . . . . . . 8  |-  ( ( e  C_  A  /\  e  ~~  b )  -> 
e  e.  { d  e.  ~P A  | 
d  ~~  b }
)
2019eximi 1582 . . . . . . 7  |-  ( E. e ( e  C_  A  /\  e  ~~  b
)  ->  E. e 
e  e.  { d  e.  ~P A  | 
d  ~~  b }
)
2112, 20syl 16 . . . . . 6  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  E. e 
e  e.  { d  e.  ~P A  | 
d  ~~  b }
)
22 eleq2 2448 . . . . . . . . 9  |-  ( { d  e.  ~P A  |  d  ~~  b }  =  { d  e. 
~P A  |  d 
~~  c }  ->  ( e  e.  { d  e.  ~P A  | 
d  ~~  b }  <->  e  e.  { d  e. 
~P A  |  d 
~~  c } ) )
2322biimpcd 216 . . . . . . . 8  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  ->  ( { d  e.  ~P A  |  d  ~~  b }  =  {
d  e.  ~P A  |  d  ~~  c }  ->  e  e.  {
d  e.  ~P A  |  d  ~~  c } ) )
2423adantl 453 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  (
b  e.  om  /\  c  e.  om )
)  /\  e  e.  { d  e.  ~P A  |  d  ~~  b } )  ->  ( {
d  e.  ~P A  |  d  ~~  b }  =  { d  e. 
~P A  |  d 
~~  c }  ->  e  e.  { d  e. 
~P A  |  d 
~~  c } ) )
2518simprbi 451 . . . . . . . . . 10  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  ->  e 
~~  b )
26 breq1 4156 . . . . . . . . . . . 12  |-  ( d  =  e  ->  (
d  ~~  c  <->  e  ~~  c ) )
2726elrab 3035 . . . . . . . . . . 11  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  c }  <->  ( e  e.  ~P A  /\  e  ~~  c ) )
2827simprbi 451 . . . . . . . . . 10  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  c }  ->  e 
~~  c )
29 ensym 7092 . . . . . . . . . . 11  |-  ( e 
~~  b  ->  b  ~~  e )
30 entr 7095 . . . . . . . . . . 11  |-  ( ( b  ~~  e  /\  e  ~~  c )  -> 
b  ~~  c )
3129, 30sylan 458 . . . . . . . . . 10  |-  ( ( e  ~~  b  /\  e  ~~  c )  -> 
b  ~~  c )
3225, 28, 31syl2an 464 . . . . . . . . 9  |-  ( ( e  e.  { d  e.  ~P A  | 
d  ~~  b }  /\  e  e.  { d  e.  ~P A  | 
d  ~~  c }
)  ->  b  ~~  c )
3332ex 424 . . . . . . . 8  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  ->  ( e  e.  { d  e.  ~P A  | 
d  ~~  c }  ->  b  ~~  c ) )
3433adantl 453 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  (
b  e.  om  /\  c  e.  om )
)  /\  e  e.  { d  e.  ~P A  |  d  ~~  b } )  ->  ( e  e.  { d  e.  ~P A  |  d  ~~  c }  ->  b  ~~  c ) )
35 nneneq 7226 . . . . . . . . 9  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  ~~  c  <->  b  =  c ) )
3635biimpd 199 . . . . . . . 8  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  ~~  c  ->  b  =  c ) )
3736ad2antlr 708 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  (
b  e.  om  /\  c  e.  om )
)  /\  e  e.  { d  e.  ~P A  |  d  ~~  b } )  ->  ( b  ~~  c  ->  b  =  c ) )
3824, 34, 373syld 53 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  (
b  e.  om  /\  c  e.  om )
)  /\  e  e.  { d  e.  ~P A  |  d  ~~  b } )  ->  ( {
d  e.  ~P A  |  d  ~~  b }  =  { d  e. 
~P A  |  d 
~~  c }  ->  b  =  c ) )
3921, 38exlimddv 1645 . . . . 5  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  ( { d  e.  ~P A  |  d  ~~  b }  =  {
d  e.  ~P A  |  d  ~~  c }  ->  b  =  c ) )
40 breq2 4157 . . . . . 6  |-  ( b  =  c  ->  (
d  ~~  b  <->  d  ~~  c ) )
4140rabbidv 2891 . . . . 5  |-  ( b  =  c  ->  { d  e.  ~P A  | 
d  ~~  b }  =  { d  e.  ~P A  |  d  ~~  c } )
4239, 41impbid1 195 . . . 4  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  ( { d  e.  ~P A  |  d  ~~  b }  =  {
d  e.  ~P A  |  d  ~~  c }  <-> 
b  =  c ) )
4342ex 424 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( ( b  e.  om  /\  c  e.  om )  ->  ( { d  e.  ~P A  |  d  ~~  b }  =  {
d  e.  ~P A  |  d  ~~  c }  <-> 
b  =  c ) ) )
449, 43dom2d 7084 . 2  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( ~P ~P A  e.  _V  ->  om  ~<_  ~P ~P A ) )
454, 44mpd 15 1  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  om  ~<_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {crab 2653   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   class class class wbr 4153   omcom 4785    ~~ cen 7042    ~<_ cdom 7043   Fincfn 7045
This theorem is referenced by:  fineqv  7260  isfin1-2  8198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-fin 7049
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