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Theorem fineqvlem 7523
Description: Lemma for fineqv 7524. (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 4473 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 462 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ~P A  e. 
_V )
3 pwexg 4473 . . 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 3434 . . . . 5  |-  { d  e.  ~P A  | 
d  ~~  b }  C_ 
~P A
6 elpw2g 4452 . . . . . 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 7522 . . . . . . . . 9  |-  ( -.  A  e.  Fin  ->  A. b  e.  om  E. e ( e  C_  A  /\  e  ~~  b
) )
1110r19.21bi 2812 . . . . . . . 8  |-  ( ( -.  A  e.  Fin  /\  b  e.  om )  ->  E. e ( e 
C_  A  /\  e  ~~  b ) )
1211ad2ant2lr 742 . . . . . . 7  |-  ( ( ( A  e.  V  /\  -.  A  e.  Fin )  /\  ( b  e. 
om  /\  c  e.  om ) )  ->  E. e
( e  C_  A  /\  e  ~~  b ) )
13 selpw 3864 . . . . . . . . . . 11  |-  ( e  e.  ~P A  <->  e  C_  A )
1413biimpri 206 . . . . . . . . . 10  |-  ( e 
C_  A  ->  e  e.  ~P A )
1514anim1i 565 . . . . . . . . 9  |-  ( ( e  C_  A  /\  e  ~~  b )  -> 
( e  e.  ~P A  /\  e  ~~  b
) )
16 breq1 4292 . . . . . . . . . 10  |-  ( d  =  e  ->  (
d  ~~  b  <->  e  ~~  b ) )
1716elrab 3114 . . . . . . . . 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 1630 . . . . . . 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 2502 . . . . . . . . 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 463 . . . . . . 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 461 . . . . . . . . . 10  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  b }  ->  e 
~~  b )
25 breq1 4292 . . . . . . . . . . . 12  |-  ( d  =  e  ->  (
d  ~~  c  <->  e  ~~  c ) )
2625elrab 3114 . . . . . . . . . . 11  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  c }  <->  ( e  e.  ~P A  /\  e  ~~  c ) )
2726simprbi 461 . . . . . . . . . 10  |-  ( e  e.  { d  e. 
~P A  |  d 
~~  c }  ->  e 
~~  c )
28 ensym 7354 . . . . . . . . . . 11  |-  ( e 
~~  b  ->  b  ~~  e )
29 entr 7357 . . . . . . . . . . 11  |-  ( ( b  ~~  e  /\  e  ~~  c )  -> 
b  ~~  c )
3028, 29sylan 468 . . . . . . . . . 10  |-  ( ( e  ~~  b  /\  e  ~~  c )  -> 
b  ~~  c )
3124, 27, 30syl2an 474 . . . . . . . . 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 463 . . . . . . 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 7490 . . . . . . . . 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 721 . . . . . . 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 1697 . . . . 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 4293 . . . . . 6  |-  ( b  =  c  ->  (
d  ~~  b  <->  d  ~~  c ) )
4039rabbidv 2962 . . . . 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 7346 . 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 1364   E.wex 1591    e. wcel 1761   {crab 2717   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857   class class class wbr 4289   omcom 6475    ~~ cen 7303    ~<_ cdom 7304   Fincfn 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-om 6476  df-er 7097  df-en 7307  df-dom 7308  df-fin 7310
This theorem is referenced by:  fineqv  7524  isfin1-2  8550
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