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Theorem fineqvlem 7734
Description: Lemma for fineqv 7735. (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 4631 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 465 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ~P A  e. 
_V )
3 pwexg 4631 . . 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 3585 . . . . 5  |-  { d  e.  ~P A  | 
d  ~~  b }  C_ 
~P A
6 elpw2g 4610 . . . . . 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 233 . . . 4  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  { d  e. 
~P A  |  d 
~~  b }  e.  ~P ~P A )
98a1d 25 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( b  e. 
om  ->  { d  e. 
~P A  |  d 
~~  b }  e.  ~P ~P A ) )
10 isinf 7733 . . . . . . . . 9  |-  ( -.  A  e.  Fin  ->  A. b  e.  om  E. e ( e  C_  A  /\  e  ~~  b
) )
1110r19.21bi 2833 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  b  e.  om )  ->  E. e ( e 
C_  A  /\  e  ~~  b ) )
1211ad2ant2lr 747 . . . . . . 7  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  E. e
( e  C_  A  /\  e  ~~  b ) )
13 selpw 4017 . . . . . . . . . . 11  |-  ( e  e.  ~P A  <->  e  C_  A )
1413biimpri 206 . . . . . . . . . 10  |-  ( e 
C_  A  ->  e  e.  ~P A )
1514anim1i 568 . . . . . . . . 9  |-  ( ( e  C_  A  /\  e  ~~  b )  -> 
( e  e.  ~P A  /\  e  ~~  b
) )
16 breq1 4450 . . . . . . . . . 10  |-  ( d  =  e  ->  (
d  ~~  b  <->  e  ~~  b ) )
1716elrab 3261 . . . . . . . . 9  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  <->  ( e  e.  ~P A  /\  e  ~~  b ) )
1815, 17sylibr 212 . . . . . . . 8  |-  ( ( e  C_  A  /\  e  ~~  b )  -> 
e  e.  { d  e.  ~P A  | 
d  ~~  b }
)
1918eximi 1635 . . . . . . 7  |-  ( E. e ( e  C_  A  /\  e  ~~  b
)  ->  E. e 
e  e.  { d  e.  ~P A  | 
d  ~~  b }
)
2012, 19syl 16 . . . . . 6  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  E. e 
e  e.  { d  e.  ~P A  | 
d  ~~  b }
)
21 eleq2 2540 . . . . . . . . 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 } ) )
2221biimpcd 224 . . . . . . . 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 } ) )
2322adantl 466 . . . . . . 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 } ) )
2417simprbi 464 . . . . . . . . . 10  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  ->  e 
~~  b )
25 breq1 4450 . . . . . . . . . . . 12  |-  ( d  =  e  ->  (
d  ~~  c  <->  e  ~~  c ) )
2625elrab 3261 . . . . . . . . . . 11  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  c }  <->  ( e  e.  ~P A  /\  e  ~~  c ) )
2726simprbi 464 . . . . . . . . . 10  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  c }  ->  e 
~~  c )
28 ensym 7564 . . . . . . . . . . 11  |-  ( e 
~~  b  ->  b  ~~  e )
29 entr 7567 . . . . . . . . . . 11  |-  ( ( b  ~~  e  /\  e  ~~  c )  -> 
b  ~~  c )
3028, 29sylan 471 . . . . . . . . . 10  |-  ( ( e  ~~  b  /\  e  ~~  c )  -> 
b  ~~  c )
3124, 27, 30syl2an 477 . . . . . . . . 9  |-  ( ( e  e.  { d  e.  ~P A  | 
d  ~~  b }  /\  e  e.  { d  e.  ~P A  | 
d  ~~  c }
)  ->  b  ~~  c )
3231ex 434 . . . . . . . 8  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  ->  ( e  e.  { d  e.  ~P A  | 
d  ~~  c }  ->  b  ~~  c ) )
3332adantl 466 . . . . . . 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 ) )
34 nneneq 7700 . . . . . . . . 9  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  ~~  c  <->  b  =  c ) )
3534biimpd 207 . . . . . . . 8  |-  ( ( b  e.  om  /\  c  e.  om )  ->  ( b  ~~  c  ->  b  =  c ) )
3635ad2antlr 726 . . . . . . 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 ) )
3723, 33, 363syld 55 . . . . . 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 ) )
3820, 37exlimddv 1702 . . . . 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 ) )
39 breq2 4451 . . . . . 6  |-  ( b  =  c  ->  (
d  ~~  b  <->  d  ~~  c ) )
4039rabbidv 3105 . . . . 5  |-  ( b  =  c  ->  { d  e.  ~P A  | 
d  ~~  b }  =  { d  e.  ~P A  |  d  ~~  c } )
4138, 40impbid1 203 . . . 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 ) )
4241ex 434 . . 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 ) ) )
439, 42dom2d 7556 . 2  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ( ~P ~P A  e.  _V  ->  om  ~<_  ~P ~P A ) )
444, 43mpd 15 1  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  om  ~<_  ~P ~P A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   {crab 2818   _Vcvv 3113    C_ wss 3476   ~Pcpw 4010   class class class wbr 4447   omcom 6684    ~~ cen 7513    ~<_ cdom 7514   Fincfn 7516
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6685  df-er 7311  df-en 7517  df-dom 7518  df-fin 7520
This theorem is referenced by:  fineqv  7735  isfin1-2  8765
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