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Theorem rankc2 8280
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 4673 . . . . 5  |-  A  C_  ~P U. A
2 rankr1b.1 . . . . . . . 8  |-  A  e. 
_V
32uniex 6573 . . . . . . 7  |-  U. A  e.  _V
43pwex 4625 . . . . . 6  |-  ~P U. A  e.  _V
54rankss 8258 . . . . 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 8252 . . . 4  |-  ( rank `  ~P U. A )  =  suc  ( rank `  U. A )
86, 7sseqtri 3531 . . 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 8248 . . . . 5  |-  ( x  e.  A  ->  ( rank `  x )  e.  ( rank `  A
) )
11 eleq1 2534 . . . . 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 2944 . . 3  |-  ( E. x  e.  A  (
rank `  x )  =  ( rank `  U. A )  ->  ( rank `  U. A )  e.  ( rank `  A
) )
14 rankon 8204 . . . 4  |-  ( rank `  U. A )  e.  On
15 rankon 8204 . . . 4  |-  ( rank `  A )  e.  On
1614, 15onsucssi 6649 . . 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 3516 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 1374    e. wcel 1762   E.wrex 2810   _Vcvv 3108    C_ wss 3471   ~Pcpw 4005   U.cuni 4240   suc csuc 4875   ` cfv 5581   rankcrnk 8172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-reg 8009  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6674  df-recs 7034  df-rdg 7068  df-r1 8173  df-rank 8174
This theorem is referenced by: (None)
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