Theorem cdacomen 9003

Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdacomen	|- ( A +c B ) ~~ ( B +c A )

Proof of Theorem cdacomen

Step	Hyp	Ref	Expression
1		1on 7567	. . . . 5 |- 1o e. On
2		xpsneng 8045	. . . . 5 |- ( ( A e. _V /\ 1o e. On ) -> ( A X. { 1o } ) ~~ A )
3	1, 2	mpan2 707	. . . 4 |- ( A e. _V -> ( A X. { 1o } ) ~~ A )
4		0ex 4790	. . . . 5 |- (/) e. _V
5		xpsneng 8045	. . . . 5 |- ( ( B e. _V /\ (/) e. _V ) -> ( B X. { (/) } ) ~~ B )
6	4, 5	mpan2 707	. . . 4 |- ( B e. _V -> ( B X. { (/) } ) ~~ B )
7		ensym 8005	. . . . 5 |- ( ( A X. { 1o } ) ~~ A -> A ~~ ( A X. { 1o } ) )
8		ensym 8005	. . . . 5 |- ( ( B X. { (/) } ) ~~ B -> B ~~ ( B X. { (/) } ) )
9		incom 3805	. . . . . . 7 |- ( ( A X. { 1o } ) i^i ( B X. { (/) } ) ) = ( ( B X. { (/) } ) i^i ( A X. { 1o } ) )
10		xp01disj 7576	. . . . . . 7 |- ( ( B X. { (/) } ) i^i ( A X. { 1o } ) ) = (/)
11	9, 10	eqtri 2644	. . . . . 6 |- ( ( A X. { 1o } ) i^i ( B X. { (/) } ) ) = (/)
12		cdaenun 8996	. . . . . 6 |- ( ( A ~~ ( A X. { 1o } ) /\ B ~~ ( B X. { (/) } ) /\ ( ( A X. { 1o } ) i^i ( B X. { (/) } ) ) = (/) ) -> ( A +c B ) ~~ ( ( A X. { 1o } ) u. ( B X. { (/) } ) ) )
13	11, 12	mp3an3 1413	. . . . 5 |- ( ( A ~~ ( A X. { 1o } ) /\ B ~~ ( B X. { (/) } ) ) -> ( A +c B ) ~~ ( ( A X. { 1o } ) u. ( B X. { (/) } ) ) )
14	7, 8, 13	syl2an 494	. . . 4 |- ( ( ( A X. { 1o } ) ~~ A /\ ( B X. { (/) } ) ~~ B ) -> ( A +c B ) ~~ ( ( A X. { 1o } ) u. ( B X. { (/) } ) ) )
15	3, 6, 14	syl2an 494	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( A +c B ) ~~ ( ( A X. { 1o } ) u. ( B X. { (/) } ) ) )
16		cdaval 8992	. . . . 5 |- ( ( B e. _V /\ A e. _V ) -> ( B +c A ) = ( ( B X. { (/) } ) u. ( A X. { 1o } ) ) )
17	16	ancoms 469	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> ( B +c A ) = ( ( B X. { (/) } ) u. ( A X. { 1o } ) ) )
18		uncom 3757	. . . 4 |- ( ( B X. { (/) } ) u. ( A X. { 1o } ) ) = ( ( A X. { 1o } ) u. ( B X. { (/) } ) )
19	17, 18	syl6eq 2672	. . 3 |- ( ( A e. _V /\ B e. _V ) -> ( B +c A ) = ( ( A X. { 1o } ) u. ( B X. { (/) } ) ) )
20	15, 19	breqtrrd 4681	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A +c B ) ~~ ( B +c A ) )
21	4	enref 7988	. . . 4 |- (/) ~~ (/)
22	21	a1i 11	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> (/) ~~ (/) )
23		cdafn 8991	. . . . 5 |- +c Fn ( _V X. _V )
24		fndm 5990	. . . . 5 |- ( +c Fn ( _V X. _V ) -> dom +c = ( _V X. _V ) )
25	23, 24	ax-mp 5	. . . 4 |- dom +c = ( _V X. _V )
26	25	ndmov 6818	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> ( A +c B ) = (/) )
27		ancom 466	. . . 4 |- ( ( A e. _V /\ B e. _V ) <-> ( B e. _V /\ A e. _V ) )
28	25	ndmov 6818	. . . 4 |- ( -. ( B e. _V /\ A e. _V ) -> ( B +c A ) = (/) )
29	27, 28	sylnbi 320	. . 3 |- ( -. ( A e. _V /\ B e. _V ) -> ( B +c A ) = (/) )
30	22, 26, 29	3brtr4d 4685	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> ( A +c B ) ~~ ( B +c A ) )
31	20, 30	pm2.61i 176	1 |- ( A +c B ) ~~ ( B +c A )

Syntax hints:   -. wn 3   /\ wa 384   = 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   dom cdm 5114   Oncon0 5723   Fn wfn 5883  (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-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-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-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by:  cdadom2  9009  cdalepw  9018  infcda  9030  alephadd  9399  gchdomtri  9451  pwxpndom  9488  gchpwdom  9492  gchhar  9501