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Theorem fineqvlem 7527
Description: Lemma for fineqv 7528. (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 4476 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 465 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ~P A  e. 
_V )
3 pwexg 4476 . . 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 3437 . . . . 5  |-  { d  e.  ~P A  | 
d  ~~  b }  C_ 
~P A
6 elpw2g 4455 . . . . . 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 7526 . . . . . . . . 9  |-  ( -.  A  e.  Fin  ->  A. b  e.  om  E. e ( e  C_  A  /\  e  ~~  b
) )
1110r19.21bi 2814 . . . . . . . 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 3867 . . . . . . . . . . 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 4295 . . . . . . . . . 10  |-  ( d  =  e  ->  (
d  ~~  b  <->  e  ~~  b ) )
1716elrab 3117 . . . . . . . . 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 1625 . . . . . . 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 2504 . . . . . . . . 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 4295 . . . . . . . . . . . 12  |-  ( d  =  e  ->  (
d  ~~  c  <->  e  ~~  c ) )
2625elrab 3117 . . . . . . . . . . 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 7358 . . . . . . . . . . 11  |-  ( e 
~~  b  ->  b  ~~  e )
29 entr 7361 . . . . . . . . . . 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 7494 . . . . . . . . 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 1692 . . . . 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 4296 . . . . . 6  |-  ( b  =  c  ->  (
d  ~~  b  <->  d  ~~  c ) )
4039rabbidv 2964 . . . . 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 7350 . 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 1369   E.wex 1586    e. wcel 1756   {crab 2719   _Vcvv 2972    C_ wss 3328   ~Pcpw 3860   class class class wbr 4292   omcom 6476    ~~ cen 7307    ~<_ cdom 7308   Fincfn 7310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-om 6477  df-er 7101  df-en 7311  df-dom 7312  df-fin 7314
This theorem is referenced by:  fineqv  7528  isfin1-2  8554
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