Theorem omxpen 8062

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.

Step	Hyp	Ref	Expression
1		xpcomeng 8052	. . 3 |- ( ( A e. On /\ B e. On ) -> ( A X. B ) ~~ ( B X. A ) )
2		xpexg 6960	. . . . 5 |- ( ( B e. On /\ A e. On ) -> ( B X. A ) e. _V )
3	2	ancoms 469	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( B X. A ) e. _V )
4		omcl 7616	. . . 4 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) e. On )
5		eqid 2622	. . . . 5 |- ( x e. B , y e. A |-> ( ( A .o x ) +o y ) ) = ( x e. B , y e. A |-> ( ( A .o x ) +o y ) )
6	5	omxpenlem 8061	. . . 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 7972	. . . 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 ) )
8	3, 4, 6, 7	syl3anc 1326	. . 3 |- ( ( A e. On /\ B e. On ) -> ( B X. A ) ~~ ( A .o B ) )
9		entr 8008	. . 3 |- ( ( ( A X. B ) ~~ ( B X. A ) /\ ( B X. A ) ~~ ( A .o B ) ) -> ( A X. B ) ~~ ( A .o B ) )
10	1, 8, 9	syl2anc 693	. 2 |- ( ( A e. On /\ B e. On ) -> ( A X. B ) ~~ ( A .o B ) )
11	10	ensymd 8007	1 |- ( ( A e. On /\ B e. On ) -> ( A .o B ) ~~ ( A X. B ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   X. cxp 5112   Oncon0 5723   -1-1-onto->wf1o 5887  (class class class)co 6650   |-> cmpt2 6652   +o coa 7557   .o comu 7558   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-en 7956

This theorem is referenced by:  xpnum  8777  infxpenc2  8845