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Theorem fineqvlem 7315
Description: Lemma for fineqv 7316. (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 4375 . . . 4  |-  ( A  e.  V  ->  ~P A  e.  _V )
21adantr 452 . . 3  |-  ( ( A  e.  V  /\  -.  A  e.  Fin )  ->  ~P A  e. 
_V )
3 pwexg 4375 . . 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 3420 . . . . 5  |-  { d  e.  ~P A  | 
d  ~~  b }  C_ 
~P A
6 elpw2g 4355 . . . . . 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 7314 . . . . . . . . 9  |-  ( -.  A  e.  Fin  ->  A. b  e.  om  E. e ( e  C_  A  /\  e  ~~  b
) )
1110r19.21bi 2796 . . . . . . . 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 2951 . . . . . . . . . . . 12  |-  e  e. 
_V
1413elpw 3797 . . . . . . . . . . 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 4207 . . . . . . . . . 10  |-  ( d  =  e  ->  (
d  ~~  b  <->  e  ~~  b ) )
1817elrab 3084 . . . . . . . . 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 1585 . . . . . . 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 2496 . . . . . . . . 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 4207 . . . . . . . . . . . 12  |-  ( d  =  e  ->  (
d  ~~  c  <->  e  ~~  c ) )
2726elrab 3084 . . . . . . . . . . 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 7148 . . . . . . . . . . 11  |-  ( e 
~~  b  ->  b  ~~  e )
30 entr 7151 . . . . . . . . . . 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 7282 . . . . . . . . 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 1648 . . . . 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 4208 . . . . . 6  |-  ( b  =  c  ->  (
d  ~~  b  <->  d  ~~  c ) )
4140rabbidv 2940 . . . . 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 7140 . 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 1550    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   class class class wbr 4204   omcom 4837    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101
This theorem is referenced by:  fineqv  7316  isfin1-2  8257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-fin 7105
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