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Theorem rankc2 8077
Description: A relationship that can be used for computation of rank. (Contributed by NM, 16-Sep-2006.)
Hypothesis
Ref Expression
rankr1b.1  |-  A  e. 
_V
Assertion
Ref Expression
rankc2  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  U. A )  ->  ( rank `  A )  =  suc  ( rank `  U. A ) )
Distinct variable group:    x, A

Proof of Theorem rankc2
StepHypRef Expression
1 pwuni 4522 . . . . 5  |-  A  C_  ~P U. A
2 rankr1b.1 . . . . . . . 8  |-  A  e. 
_V
32uniex 6375 . . . . . . 7  |-  U. A  e.  _V
43pwex 4474 . . . . . 6  |-  ~P U. A  e.  _V
54rankss 8055 . . . . 5  |-  ( A 
C_  ~P U. A  -> 
( rank `  A )  C_  ( rank `  ~P U. A ) )
61, 5ax-mp 5 . . . 4  |-  ( rank `  A )  C_  ( rank `  ~P U. A
)
73rankpw 8049 . . . 4  |-  ( rank `  ~P U. A )  =  suc  ( rank `  U. A )
86, 7sseqtri 3387 . . 3  |-  ( rank `  A )  C_  suc  ( rank `  U. A )
98a1i 11 . 2  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  U. A )  ->  ( rank `  A )  C_  suc  ( rank `  U. A ) )
102rankel 8045 . . . . 5  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
11 eleq1 2502 . . . . 5  |-  ( (
rank `  x )  =  ( rank `  U. A )  ->  (
( rank `  x )  e.  ( rank `  A
)  <->  ( rank `  U. A )  e.  (
rank `  A )
) )
1210, 11syl5ibcom 220 . . . 4  |-  ( x  e.  A  ->  (
( rank `  x )  =  ( rank `  U. A )  ->  ( rank `  U. A )  e.  ( rank `  A
) ) )
1312rexlimiv 2834 . . 3  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  U. A )  ->  ( rank `  U. A )  e.  ( rank `  A
) )
14 rankon 8001 . . . 4  |-  ( rank `  U. A )  e.  On
15 rankon 8001 . . . 4  |-  ( rank `  A )  e.  On
1614, 15onsucssi 6451 . . 3  |-  ( (
rank `  U. A )  e.  ( rank `  A
)  <->  suc  ( rank `  U. A )  C_  ( rank `  A ) )
1713, 16sylib 196 . 2  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  U. A )  ->  suc  ( rank `  U. A ) 
C_  ( rank `  A
) )
189, 17eqssd 3372 1  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  U. A )  ->  ( rank `  A )  =  suc  ( rank `  U. A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   E.wrex 2715   _Vcvv 2971    C_ wss 3327   ~Pcpw 3859   U.cuni 4090   suc csuc 4720   ` cfv 5417   rankcrnk 7969
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 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-reg 7806  ax-inf2 7846
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 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-om 6476  df-recs 6831  df-rdg 6865  df-r1 7970  df-rank 7971
This theorem is referenced by: (None)
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