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Theorem rankmapu 8085
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 7248 . . 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 6508 . . . . 5  |-  ( B  X.  A )  e. 
_V
54pwex 4475 . . . 4  |-  ~P ( B  X.  A )  e. 
_V
65rankss 8056 . . 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 8050 . . 3  |-  ( rank `  ~P ( B  X.  A ) )  =  suc  ( rank `  ( B  X.  A ) )
92, 3rankxpu 8083 . . . . 5  |-  ( rank `  ( B  X.  A
) )  C_  suc  suc  ( rank `  ( B  u.  A )
)
10 uncom 3500 . . . . . . . 8  |-  ( B  u.  A )  =  ( A  u.  B
)
1110fveq2i 5694 . . . . . . 7  |-  ( rank `  ( B  u.  A
) )  =  (
rank `  ( A  u.  B ) )
12 suceq 4784 . . . . . . 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 4784 . . . . . 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 3388 . . . 4  |-  ( rank `  ( B  X.  A
) )  C_  suc  suc  ( rank `  ( A  u.  B )
)
17 rankon 8002 . . . . . 6  |-  ( rank `  ( B  X.  A
) )  e.  On
1817onordi 4823 . . . . 5  |-  Ord  ( rank `  ( B  X.  A ) )
19 rankon 8002 . . . . . . . 8  |-  ( rank `  ( A  u.  B
) )  e.  On
2019onsuci 6449 . . . . . . 7  |-  suc  ( rank `  ( A  u.  B ) )  e.  On
2120onsuci 6449 . . . . . 6  |-  suc  suc  ( rank `  ( A  u.  B ) )  e.  On
2221onordi 4823 . . . . 5  |-  Ord  suc  suc  ( rank `  ( A  u.  B )
)
23 ordsucsssuc 6434 . . . . 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 3386 . 2  |-  ( rank `  ~P ( B  X.  A ) )  C_  suc  suc  suc  ( rank `  ( A  u.  B
) )
277, 26sstri 3365 1  |-  ( rank `  ( A  ^m  B
) )  C_  suc  suc 
suc  ( rank `  ( A  u.  B )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369    e. wcel 1756   _Vcvv 2972    u. cun 3326    C_ wss 3328   ~Pcpw 3860   Ord word 4718   suc csuc 4721    X. cxp 4838   ` cfv 5418  (class class class)co 6091    ^m cmap 7214   rankcrnk 7970
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-reg 7807  ax-inf2 7847
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-map 7216  df-pm 7217  df-r1 7971  df-rank 7972
This theorem is referenced by: (None)
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