Theorem cdaun 8994

Description: Cardinal addition is equinumerous to union for disjoint sets. (Contributed by NM, 5-Apr-2007.)

Assertion

Ref	Expression
cdaun	|- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A +c B ) ~~ ( A u. B ) )

Proof of Theorem cdaun

Step	Hyp	Ref	Expression
1		cdaval 8992	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
2	1	3adant3 1081	. 2 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A +c B ) = ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) )
3		0ex 4790	. . . . . 6 |- (/) e. _V
4		xpsneng 8045	. . . . . 6 |- ( ( A e. V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
5	3, 4	mpan2 707	. . . . 5 |- ( A e. V -> ( A X. { (/) } ) ~~ A )
6		1on 7567	. . . . . 6 |- 1o e. On
7		xpsneng 8045	. . . . . 6 |- ( ( B e. W /\ 1o e. On ) -> ( B X. { 1o } ) ~~ B )
8	6, 7	mpan2 707	. . . . 5 |- ( B e. W -> ( B X. { 1o } ) ~~ B )
9	5, 8	anim12i 590	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) )
10		xp01disj 7576	. . . . 5 |- ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/)
11	10	jctl 564	. . . 4 |- ( ( A i^i B ) = (/) -> ( ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) /\ ( A i^i B ) = (/) ) )
12		unen 8040	. . . 4 |- ( ( ( ( A X. { (/) } ) ~~ A /\ ( B X. { 1o } ) ~~ B ) /\ ( ( ( A X. { (/) } ) i^i ( B X. { 1o } ) ) = (/) /\ ( A i^i B ) = (/) ) ) -> ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) ~~ ( A u. B ) )
13	9, 11, 12	syl2an 494	. . 3 |- ( ( ( A e. V /\ B e. W ) /\ ( A i^i B ) = (/) ) -> ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) ~~ ( A u. B ) )
14	13	3impa 1259	. 2 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( ( A X. { (/) } ) u. ( B X. { 1o } ) ) ~~ ( A u. B ) )
15	2, 14	eqbrtrd 4675	1 |- ( ( A e. V /\ B e. W /\ ( A i^i B ) = (/) ) -> ( A +c B ) ~~ ( A u. B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   Oncon0 5723  (class class class)co 6650   1oc1o 7553   ~~ cen 7952   +c ccda 8989

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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-ord 5726  df-on 5727  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-1o 7560  df-en 7956  df-cda 8990

This theorem is referenced by:  cdaenun  8996  cda0en  9001  ficardun  9024  ackbij1lem9  9050  canthp1lem1  9474