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Theorem rankmapu 8313
Description: An upper bound on the rank of set exponentiation. (Contributed by Gérard Lang, 5-Aug-2018.)
Hypotheses
Ref Expression
rankxpl.1  |-  A  e. 
_V
rankxpl.2  |-  B  e. 
_V
Assertion
Ref Expression
rankmapu  |-  ( rank `  ( A  ^m  B
) )  C_  suc  suc 
suc  ( rank `  ( A  u.  B )
)

Proof of Theorem rankmapu
StepHypRef Expression
1 mapsspw 7473 . . 3  |-  ( A  ^m  B )  C_  ~P ( B  X.  A
)
2 rankxpl.2 . . . . . 6  |-  B  e. 
_V
3 rankxpl.1 . . . . . 6  |-  A  e. 
_V
42, 3xpex 6603 . . . . 5  |-  ( B  X.  A )  e. 
_V
54pwex 4639 . . . 4  |-  ~P ( B  X.  A )  e. 
_V
65rankss 8284 . . 3  |-  ( ( A  ^m  B ) 
C_  ~P ( B  X.  A )  ->  ( rank `  ( A  ^m  B ) )  C_  ( rank `  ~P ( B  X.  A ) ) )
71, 6ax-mp 5 . 2  |-  ( rank `  ( A  ^m  B
) )  C_  ( rank `  ~P ( B  X.  A ) )
84rankpw 8278 . . 3  |-  ( rank `  ~P ( B  X.  A ) )  =  suc  ( rank `  ( B  X.  A ) )
92, 3rankxpu 8311 . . . . 5  |-  ( rank `  ( B  X.  A
) )  C_  suc  suc  ( rank `  ( B  u.  A )
)
10 uncom 3644 . . . . . . . 8  |-  ( B  u.  A )  =  ( A  u.  B
)
1110fveq2i 5875 . . . . . . 7  |-  ( rank `  ( B  u.  A
) )  =  (
rank `  ( A  u.  B ) )
12 suceq 4952 . . . . . . 7  |-  ( (
rank `  ( B  u.  A ) )  =  ( rank `  ( A  u.  B )
)  ->  suc  ( rank `  ( B  u.  A
) )  =  suc  ( rank `  ( A  u.  B ) ) )
1311, 12ax-mp 5 . . . . . 6  |-  suc  ( rank `  ( B  u.  A ) )  =  suc  ( rank `  ( A  u.  B )
)
14 suceq 4952 . . . . . 6  |-  ( suc  ( rank `  ( B  u.  A )
)  =  suc  ( rank `  ( A  u.  B ) )  ->  suc  suc  ( rank `  ( B  u.  A )
)  =  suc  suc  ( rank `  ( A  u.  B ) ) )
1513, 14ax-mp 5 . . . . 5  |-  suc  suc  ( rank `  ( B  u.  A ) )  =  suc  suc  ( rank `  ( A  u.  B
) )
169, 15sseqtri 3531 . . . 4  |-  ( rank `  ( B  X.  A
) )  C_  suc  suc  ( rank `  ( A  u.  B )
)
17 rankon 8230 . . . . . 6  |-  ( rank `  ( B  X.  A
) )  e.  On
1817onordi 4991 . . . . 5  |-  Ord  ( rank `  ( B  X.  A ) )
19 rankon 8230 . . . . . . . 8  |-  ( rank `  ( A  u.  B
) )  e.  On
2019onsuci 6672 . . . . . . 7  |-  suc  ( rank `  ( A  u.  B ) )  e.  On
2120onsuci 6672 . . . . . 6  |-  suc  suc  ( rank `  ( A  u.  B ) )  e.  On
2221onordi 4991 . . . . 5  |-  Ord  suc  suc  ( rank `  ( A  u.  B )
)
23 ordsucsssuc 6657 . . . . 5  |-  ( ( Ord  ( rank `  ( B  X.  A ) )  /\  Ord  suc  suc  ( rank `  ( A  u.  B ) ) )  ->  ( ( rank `  ( B  X.  A
) )  C_  suc  suc  ( rank `  ( A  u.  B )
)  <->  suc  ( rank `  ( B  X.  A ) ) 
C_  suc  suc  suc  ( rank `  ( A  u.  B ) ) ) )
2418, 22, 23mp2an 672 . . . 4  |-  ( (
rank `  ( B  X.  A ) )  C_  suc  suc  ( rank `  ( A  u.  B )
)  <->  suc  ( rank `  ( B  X.  A ) ) 
C_  suc  suc  suc  ( rank `  ( A  u.  B ) ) )
2516, 24mpbi 208 . . 3  |-  suc  ( rank `  ( B  X.  A ) )  C_  suc  suc  suc  ( rank `  ( A  u.  B
) )
268, 25eqsstri 3529 . 2  |-  ( rank `  ~P ( B  X.  A ) )  C_  suc  suc  suc  ( rank `  ( A  u.  B
) )
277, 26sstri 3508 1  |-  ( rank `  ( A  ^m  B
) )  C_  suc  suc 
suc  ( rank `  ( A  u.  B )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   ~Pcpw 4015   Ord word 4886   suc csuc 4889    X. cxp 5006   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   rankcrnk 8198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-map 7440  df-pm 7441  df-r1 8199  df-rank 8200
This theorem is referenced by: (None)
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