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Theorem omxpen 6964
Description: The cardinal and ordinal products are always equinumerous. Exercise 10 of [TakeutiZaring] p. 89. (Contributed by Mario Carneiro, 3-Mar-2013.)
Assertion
Ref Expression
omxpen  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  ~~  ( A  X.  B ) )

Proof of Theorem omxpen
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcomeng 6954 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
2 xpexg 4800 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  ( B  X.  A
)  e.  _V )
32ancoms 439 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  X.  A
)  e.  _V )
4 omcl 6535 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
5 eqid 2283 . . . . 5  |-  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )
65omxpenlem 6963 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y
) ) : ( B  X.  A ) -1-1-onto-> ( A  .o  B ) )
7 f1oen2g 6878 . . . 4  |-  ( ( ( B  X.  A
)  e.  _V  /\  ( A  .o  B
)  e.  On  /\  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y
) ) : ( B  X.  A ) -1-1-onto-> ( A  .o  B ) )  ->  ( B  X.  A )  ~~  ( A  .o  B ) )
83, 4, 6, 7syl3anc 1182 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  X.  A
)  ~~  ( A  .o  B ) )
9 entr 6913 . . 3  |-  ( ( ( A  X.  B
)  ~~  ( B  X.  A )  /\  ( B  X.  A )  ~~  ( A  .o  B
) )  ->  ( A  X.  B )  ~~  ( A  .o  B
) )
101, 8, 9syl2anc 642 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( A  .o  B ) )
11 ensym 6910 . 2  |-  ( ( A  X.  B ) 
~~  ( A  .o  B )  ->  ( A  .o  B )  ~~  ( A  X.  B
) )
1210, 11syl 15 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  ~~  ( A  X.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   Oncon0 4392    X. cxp 4687   -1-1-onto->wf1o 5254  (class class class)co 5858    e. cmpt2 5860    +o coa 6476    .o comu 6477    ~~ cen 6860
This theorem is referenced by:  xpnum  7584  infxpenc2  7649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-en 6864
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