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Theorem omxpen 6966
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 6956 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  X.  B
)  ~~  ( B  X.  A ) )
2 xpexg 4802 . . . . 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 6537 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  .o  B
)  e.  On )
5 eqid 2285 . . . . 5  |-  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )  =  ( x  e.  B ,  y  e.  A  |->  ( ( A  .o  x )  +o  y ) )
65omxpenlem 6965 . . . 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 6880 . . . 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 6915 . . 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 6912 . 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 1686   _Vcvv 2790   class class class wbr 4025   Oncon0 4394    X. cxp 4689   -1-1-onto->wf1o 5256  (class class class)co 5860    e. cmpt2 5862    +o coa 6478    .o comu 6479    ~~ cen 6862
This theorem is referenced by:  xpnum  7586  infxpenc2  7651
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-er 6662  df-en 6866
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