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Theorem cardprclem 7830
Description: Lemma for cardprc 7831. (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 2476 . . . . . . . 8  |-  ( x  e.  A  <->  x  e.  { x  |  ( card `  x )  =  x } )
3 abid 2400 . . . . . . . 8  |-  ( x  e.  { x  |  ( card `  x
)  =  x }  <->  (
card `  x )  =  x )
4 iscard 7826 . . . . . . . 8  |-  ( (
card `  x )  =  x  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
52, 3, 43bitri 263 . . . . . . 7  |-  ( x  e.  A  <->  ( x  e.  On  /\  A. y  e.  x  y  ~<  x ) )
65simplbi 447 . . . . . 6  |-  ( x  e.  A  ->  x  e.  On )
76ssriv 3320 . . . . 5  |-  A  C_  On
8 ssonuni 4734 . . . . 5  |-  ( A  e.  _V  ->  ( A  C_  On  ->  U. A  e.  On ) )
97, 8mpi 17 . . . 4  |-  ( A  e.  _V  ->  U. A  e.  On )
10 domrefg 7109 . . . . 5  |-  ( U. A  e.  On  ->  U. A  ~<_  U. A )
119, 10syl 16 . . . 4  |-  ( A  e.  _V  ->  U. A  ~<_  U. A )
12 elharval 7495 . . . 4  |-  ( U. A  e.  (har `  U. A )  <->  ( U. A  e.  On  /\  U. A  ~<_  U. A ) )
139, 11, 12sylanbrc 646 . . 3  |-  ( A  e.  _V  ->  U. A  e.  (har `  U. A ) )
147sseli 3312 . . . . . . . 8  |-  ( z  e.  A  ->  z  e.  On )
15 domrefg 7109 . . . . . . . . . 10  |-  ( z  e.  On  ->  z  ~<_  z )
1615ancli 535 . . . . . . . . 9  |-  ( z  e.  On  ->  (
z  e.  On  /\  z  ~<_  z ) )
17 elharval 7495 . . . . . . . . 9  |-  ( z  e.  (har `  z
)  <->  ( z  e.  On  /\  z  ~<_  z ) )
1816, 17sylibr 204 . . . . . . . 8  |-  ( z  e.  On  ->  z  e.  (har `  z )
)
1914, 18syl 16 . . . . . . 7  |-  ( z  e.  A  ->  z  e.  (har `  z )
)
20 harcard 7829 . . . . . . . 8  |-  ( card `  (har `  z )
)  =  (har `  z )
21 fvex 5709 . . . . . . . . 9  |-  (har `  z )  e.  _V
22 fveq2 5695 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  ( card `  x )  =  (
card `  (har `  z
) ) )
23 id 20 . . . . . . . . . 10  |-  ( x  =  (har `  z
)  ->  x  =  (har `  z ) )
2422, 23eqeq12d 2426 . . . . . . . . 9  |-  ( x  =  (har `  z
)  ->  ( ( card `  x )  =  x  <->  ( card `  (har `  z ) )  =  (har `  z )
) )
2521, 24, 1elab2 3053 . . . . . . . 8  |-  ( (har
`  z )  e.  A  <->  ( card `  (har `  z ) )  =  (har `  z )
)
2620, 25mpbir 201 . . . . . . 7  |-  (har `  z )  e.  A
27 eleq2 2473 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( z  e.  w  <->  z  e.  (har
`  z ) ) )
28 eleq1 2472 . . . . . . . . 9  |-  ( w  =  (har `  z
)  ->  ( w  e.  A  <->  (har `  z )  e.  A ) )
2927, 28anbi12d 692 . . . . . . . 8  |-  ( w  =  (har `  z
)  ->  ( (
z  e.  w  /\  w  e.  A )  <->  ( z  e.  (har `  z )  /\  (har `  z )  e.  A
) ) )
3021, 29spcev 3011 . . . . . . 7  |-  ( ( z  e.  (har `  z )  /\  (har `  z )  e.  A
)  ->  E. w
( z  e.  w  /\  w  e.  A
) )
3119, 26, 30sylancl 644 . . . . . 6  |-  ( z  e.  A  ->  E. w
( z  e.  w  /\  w  e.  A
) )
32 eluni 3986 . . . . . 6  |-  ( z  e.  U. A  <->  E. w
( z  e.  w  /\  w  e.  A
) )
3331, 32sylibr 204 . . . . 5  |-  ( z  e.  A  ->  z  e.  U. A )
3433ssriv 3320 . . . 4  |-  A  C_  U. A
35 harcard 7829 . . . . 5  |-  ( card `  (har `  U. A ) )  =  (har `  U. A )
36 fvex 5709 . . . . . 6  |-  (har `  U. A )  e.  _V
37 fveq2 5695 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  ( card `  x )  =  ( card `  (har ` 
U. A ) ) )
38 id 20 . . . . . . 7  |-  ( x  =  (har `  U. A )  ->  x  =  (har `  U. A ) )
3937, 38eqeq12d 2426 . . . . . 6  |-  ( x  =  (har `  U. A )  ->  (
( card `  x )  =  x  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) ) )
4036, 39, 1elab2 3053 . . . . 5  |-  ( (har
`  U. A )  e.  A  <->  ( card `  (har ` 
U. A ) )  =  (har `  U. A ) )
4135, 40mpbir 201 . . . 4  |-  (har `  U. A )  e.  A
4234, 41sselii 3313 . . 3  |-  (har `  U. A )  e.  U. A
4313, 42jctir 525 . 2  |-  ( A  e.  _V  ->  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
44 eloni 4559 . . 3  |-  ( U. A  e.  On  ->  Ord  U. A )
45 ordn2lp 4569 . . 3  |-  ( Ord  U. A  ->  -.  ( U. A  e.  (har ` 
U. A )  /\  (har `  U. A )  e.  U. A ) )
469, 44, 453syl 19 . 2  |-  ( A  e.  _V  ->  -.  ( U. A  e.  (har
`  U. A )  /\  (har `  U. A )  e.  U. A ) )
4743, 46pm2.65i 167 1  |-  -.  A  e.  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2398   A.wral 2674   _Vcvv 2924    C_ wss 3288   U.cuni 3983   class class class wbr 4180   Ord word 4548   Oncon0 4549   ` cfv 5421    ~<_ cdom 7074    ~< csdm 7075  harchar 7488   cardccrd 7786
This theorem is referenced by:  cardprc  7831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6516  df-recs 6600  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-oi 7443  df-har 7490  df-card 7790
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