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Theorem omxpen 6960
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 6950 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
2 xpexg 4799 . . . . 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 6531 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
5 eqid 2284 . . . . 5  |-  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )
65omxpenlem 6959 . . . 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 6874 . . . 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 6909 . . 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 6906 . 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 1685   _Vcvv 2789   class class class wbr 4024   Oncon0 4391    X. cxp 4686   -1-1-onto->wf1o 5220  (class class class)co 5820    e. cmpt2 5822    +o coa 6472    .o comu 6473    ~~ cen 6856
This theorem is referenced by:  xpnum  7580  infxpenc2  7645
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-omul 6480  df-er 6656  df-en 6860
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