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Theorem cardprclem 8141
Description: Lemma for cardprc 8142. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)
Hypothesis
Ref Expression
cardprclem.1  |-  A  =  { x  |  (
card `  x )  =  x }
Assertion
Ref Expression
cardprclem  |-  -.  A  e.  _V
Distinct variable group:    x, A

Proof of Theorem cardprclem
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cardprclem.1 . . . . . . . . 9  |-  A  =  { x  |  (
card `  x )  =  x }
21eleq2i 2502 . . . . . . . 8  |-  ( x  e.  A  <->  x  e.  { x  |  ( card `  x )  =  x } )
3 abid 2426 . . . . . . . 8  |-  ( x  e.  { x  |  ( card `  x
)  =  x }  <->  (
card `  x )  =  x )
4 iscard 8137 . . . . . . . 8  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
52, 3, 43bitri 271 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
65simplbi 460 . . . . . 6  |-  ( x  e.  A  ->  x  e.  On )
76ssriv 3355 . . . . 5  |-  A  C_  On
8 ssonuni 6393 . . . . 5  |-  ( A  e.  _V  ->  ( A  C_  On  ->  U. A  e.  On ) )
97, 8mpi 17 . . . 4  |-  ( A  e.  _V  ->  U. A  e.  On )
10 domrefg 7336 . . . . 5  |-  ( U. A  e.  On  ->  U. A  ~<_  U. A )
119, 10syl 16 . . . 4  |-  ( A  e.  _V  ->  U. A  ~<_  U. A )
12 elharval 7770 . . . 4  |-  ( U. A  e.  (har `  U. A )  <->  ( U. A  e.  On  /\  U. A  ~<_  U. A ) )
139, 11, 12sylanbrc 664 . . 3  |-  ( A  e.  _V  ->  U. A  e.  (har `  U. A ) )
147sseli 3347 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  On )
15 domrefg 7336 . . . . . . . . . 10  |-  ( z  e.  On  ->  z  ~<_  z )
1615ancli 551 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  e.  On  /\  z  ~<_  z ) )
17 elharval 7770 . . . . . . . . 9  |-  ( z  e.  (har `  z
)  <->  ( z  e.  On  /\  z  ~<_  z ) )
1816, 17sylibr 212 . . . . . . . 8  |-  ( z  e.  On  ->  z  e.  (har `  z )
)
1914, 18syl 16 . . . . . . 7  |-  ( z  e.  A  ->  z  e.  (har `  z )
)
20 harcard 8140 . . . . . . . 8  |-  ( card `  (har `  z )
)  =  (har `  z )
21 fvex 5696 . . . . . . . . 9  |-  (har `  z )  e.  _V
22 fveq2 5686 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  ( card `  x )  =  (
card `  (har `  z
) ) )
23 id 22 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  x  =  (har `  z ) )
2422, 23eqeq12d 2452 . . . . . . . . 9  |-  ( x  =  (har `  z
)  ->  ( ( card `  x )  =  x  <->  ( card `  (har `  z ) )  =  (har `  z )
) )
2521, 24, 1elab2 3104 . . . . . . . 8  |-  ( (har
`  z )  e.  A  <->  ( card `  (har `  z ) )  =  (har `  z )
)
2620, 25mpbir 209 . . . . . . 7  |-  (har `  z )  e.  A
27 eleq2 2499 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( z  e.  w  <->  z  e.  (har
`  z ) ) )
28 eleq1 2498 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( w  e.  A  <->  (har `  z )  e.  A ) )
2927, 28anbi12d 710 . . . . . . . 8  |-  ( w  =  (har `  z
)  ->  ( (
z  e.  w  /\  w  e.  A )  <->  ( z  e.  (har `  z )  /\  (har `  z )  e.  A
) ) )
3021, 29spcev 3059 . . . . . . 7  |-  ( ( z  e.  (har `  z )  /\  (har `  z )  e.  A
)  ->  E. w
( z  e.  w  /\  w  e.  A
) )
3119, 26, 30sylancl 662 . . . . . 6  |-  ( z  e.  A  ->  E. w
( z  e.  w  /\  w  e.  A
) )
32 eluni 4089 . . . . . 6  |-  ( z  e.  U. A  <->  E. w
( z  e.  w  /\  w  e.  A
) )
3331, 32sylibr 212 . . . . 5  |-  ( z  e.  A  ->  z  e.  U. A )
3433ssriv 3355 . . . 4  |-  A  C_  U. A
35 harcard 8140 . . . . 5  |-  ( card `  (har `  U. A ) )  =  (har `  U. A )
36 fvex 5696 . . . . . 6  |-  (har `  U. A )  e.  _V
37 fveq2 5686 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  ( card `  x )  =  ( card `  (har ` 
U. A ) ) )
38 id 22 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  x  =  (har `  U. A ) )
3937, 38eqeq12d 2452 . . . . . 6  |-  ( x  =  (har `  U. A )  ->  (
( card `  x )  =  x  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) ) )
4036, 39, 1elab2 3104 . . . . 5  |-  ( (har
`  U. A )  e.  A  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) )
4135, 40mpbir 209 . . . 4  |-  (har `  U. A )  e.  A
4234, 41sselii 3348 . . 3  |-  (har `  U. A )  e.  U. A
4313, 42jctir 538 . 2  |-  ( A  e.  _V  ->  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
44 eloni 4724 . . 3  |-  ( U. A  e.  On  ->  Ord  U. A )
45 ordn2lp 4734 . . 3  |-  ( Ord  U. A  ->  -.  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
469, 44, 453syl 20 . 2  |-  ( A  e.  _V  ->  -.  ( U. A  e.  (har
`  U. A )  /\  (har `  U. A )  e.  U. A ) )
4743, 46pm2.65i 173 1  |-  -.  A  e.  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424   A.wral 2710   _Vcvv 2967    C_ wss 3323   U.cuni 4086   class class class wbr 4287   Ord word 4713   Oncon0 4714   ` cfv 5413    ~<_ cdom 7300    ~< csdm 7301  harchar 7763   cardccrd 8097
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 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-recs 6824  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-oi 7716  df-har 7765  df-card 8101
This theorem is referenced by:  cardprc  8142
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