Theorem cdaassen 9004

Description: Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 26-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
cdaassen	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A +c B ) +c C ) ~~ ( A +c ( B +c C ) ) )

Proof of Theorem cdaassen

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A e. V )
2		0ex 4790	. . . . . 6 |- (/) e. _V
3		xpsneng 8045	. . . . . 6 |- ( ( A e. V /\ (/) e. _V ) -> ( A X. { (/) } ) ~~ A )
4	1, 2, 3	sylancl 694	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A X. { (/) } ) ~~ A )
5	4	ensymd 8007	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ ( A X. { (/) } ) )
6		simp2 1062	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
7		snex 4908	. . . . . . . 8 |- { (/) } e. _V
8		xpexg 6960	. . . . . . . 8 |- ( ( B e. W /\ { (/) } e. _V ) -> ( B X. { (/) } ) e. _V )
9	6, 7, 8	sylancl 694	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( B X. { (/) } ) e. _V )
10		1on 7567	. . . . . . 7 |- 1o e. On
11		xpsneng 8045	. . . . . . 7 |- ( ( ( B X. { (/) } ) e. _V /\ 1o e. On ) -> ( ( B X. { (/) } ) X. { 1o } ) ~~ ( B X. { (/) } ) )
12	9, 10, 11	sylancl 694	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( B X. { (/) } ) X. { 1o } ) ~~ ( B X. { (/) } ) )
13		xpsneng 8045	. . . . . . 7 |- ( ( B e. W /\ (/) e. _V ) -> ( B X. { (/) } ) ~~ B )
14	6, 2, 13	sylancl 694	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( B X. { (/) } ) ~~ B )
15		entr 8008	. . . . . 6 |- ( ( ( ( B X. { (/) } ) X. { 1o } ) ~~ ( B X. { (/) } ) /\ ( B X. { (/) } ) ~~ B ) -> ( ( B X. { (/) } ) X. { 1o } ) ~~ B )
16	12, 14, 15	syl2anc 693	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( B X. { (/) } ) X. { 1o } ) ~~ B )
17	16	ensymd 8007	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( ( B X. { (/) } ) X. { 1o } ) )
18		xp01disj 7576	. . . . 5 |- ( ( A X. { (/) } ) i^i ( ( B X. { (/) } ) X. { 1o } ) ) = (/)
19	18	a1i 11	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A X. { (/) } ) i^i ( ( B X. { (/) } ) X. { 1o } ) ) = (/) )
20		cdaenun 8996	. . . 4 |- ( ( A ~~ ( A X. { (/) } ) /\ B ~~ ( ( B X. { (/) } ) X. { 1o } ) /\ ( ( A X. { (/) } ) i^i ( ( B X. { (/) } ) X. { 1o } ) ) = (/) ) -> ( A +c B ) ~~ ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) )
21	5, 17, 19, 20	syl3anc 1326	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A +c B ) ~~ ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) )
22		simp3 1063	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
23		snex 4908	. . . . . . 7 |- { 1o } e. _V
24		xpexg 6960	. . . . . . 7 |- ( ( C e. X /\ { 1o } e. _V ) -> ( C X. { 1o } ) e. _V )
25	22, 23, 24	sylancl 694	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( C X. { 1o } ) e. _V )
26		xpsneng 8045	. . . . . 6 |- ( ( ( C X. { 1o } ) e. _V /\ 1o e. On ) -> ( ( C X. { 1o } ) X. { 1o } ) ~~ ( C X. { 1o } ) )
27	25, 10, 26	sylancl 694	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( C X. { 1o } ) X. { 1o } ) ~~ ( C X. { 1o } ) )
28		xpsneng 8045	. . . . . 6 |- ( ( C e. X /\ 1o e. On ) -> ( C X. { 1o } ) ~~ C )
29	22, 10, 28	sylancl 694	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( C X. { 1o } ) ~~ C )
30		entr 8008	. . . . 5 |- ( ( ( ( C X. { 1o } ) X. { 1o } ) ~~ ( C X. { 1o } ) /\ ( C X. { 1o } ) ~~ C ) -> ( ( C X. { 1o } ) X. { 1o } ) ~~ C )
31	27, 29, 30	syl2anc 693	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( C X. { 1o } ) X. { 1o } ) ~~ C )
32	31	ensymd 8007	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( ( C X. { 1o } ) X. { 1o } ) )
33		indir 3875	. . . . 5 |- ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) = ( ( ( A X. { (/) } ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) u. ( ( ( B X. { (/) } ) X. { 1o } ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) )
34		xp01disj 7576	. . . . . 6 |- ( ( A X. { (/) } ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) = (/)
35		xp01disj 7576	. . . . . . . 8 |- ( ( B X. { (/) } ) i^i ( C X. { 1o } ) ) = (/)
36	35	xpeq1i 5135	. . . . . . 7 |- ( ( ( B X. { (/) } ) i^i ( C X. { 1o } ) ) X. { 1o } ) = ( (/) X. { 1o } )
37		xpindir 5256	. . . . . . 7 |- ( ( ( B X. { (/) } ) i^i ( C X. { 1o } ) ) X. { 1o } ) = ( ( ( B X. { (/) } ) X. { 1o } ) i^i ( ( C X. { 1o } ) X. { 1o } ) )
38		0xp 5199	. . . . . . 7 |- ( (/) X. { 1o } ) = (/)
39	36, 37, 38	3eqtr3i 2652	. . . . . 6 |- ( ( ( B X. { (/) } ) X. { 1o } ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) = (/)
40	34, 39	uneq12i 3765	. . . . 5 |- ( ( ( A X. { (/) } ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) u. ( ( ( B X. { (/) } ) X. { 1o } ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) ) = ( (/) u. (/) )
41		un0 3967	. . . . 5 |- ( (/) u. (/) ) = (/)
42	33, 40, 41	3eqtri 2648	. . . 4 |- ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) = (/)
43	42	a1i 11	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) = (/) )
44		cdaenun 8996	. . 3 |- ( ( ( A +c B ) ~~ ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) /\ C ~~ ( ( C X. { 1o } ) X. { 1o } ) /\ ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) i^i ( ( C X. { 1o } ) X. { 1o } ) ) = (/) ) -> ( ( A +c B ) +c C ) ~~ ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
45	21, 32, 43, 44	syl3anc 1326	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A +c B ) +c C ) ~~ ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
46		ovex 6678	. . . . 5 |- ( B +c C ) e. _V
47		cdaval 8992	. . . . 5 |- ( ( A e. V /\ ( B +c C ) e. _V ) -> ( A +c ( B +c C ) ) = ( ( A X. { (/) } ) u. ( ( B +c C ) X. { 1o } ) ) )
48	46, 47	mpan2 707	. . . 4 |- ( A e. V -> ( A +c ( B +c C ) ) = ( ( A X. { (/) } ) u. ( ( B +c C ) X. { 1o } ) ) )
49		cdaval 8992	. . . . . . . 8 |- ( ( B e. W /\ C e. X ) -> ( B +c C ) = ( ( B X. { (/) } ) u. ( C X. { 1o } ) ) )
50	49	xpeq1d 5138	. . . . . . 7 |- ( ( B e. W /\ C e. X ) -> ( ( B +c C ) X. { 1o } ) = ( ( ( B X. { (/) } ) u. ( C X. { 1o } ) ) X. { 1o } ) )
51		xpundir 5172	. . . . . . 7 |- ( ( ( B X. { (/) } ) u. ( C X. { 1o } ) ) X. { 1o } ) = ( ( ( B X. { (/) } ) X. { 1o } ) u. ( ( C X. { 1o } ) X. { 1o } ) )
52	50, 51	syl6eq 2672	. . . . . 6 |- ( ( B e. W /\ C e. X ) -> ( ( B +c C ) X. { 1o } ) = ( ( ( B X. { (/) } ) X. { 1o } ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
53	52	uneq2d 3767	. . . . 5 |- ( ( B e. W /\ C e. X ) -> ( ( A X. { (/) } ) u. ( ( B +c C ) X. { 1o } ) ) = ( ( A X. { (/) } ) u. ( ( ( B X. { (/) } ) X. { 1o } ) u. ( ( C X. { 1o } ) X. { 1o } ) ) ) )
54		unass 3770	. . . . 5 |- ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) u. ( ( C X. { 1o } ) X. { 1o } ) ) = ( ( A X. { (/) } ) u. ( ( ( B X. { (/) } ) X. { 1o } ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
55	53, 54	syl6eqr 2674	. . . 4 |- ( ( B e. W /\ C e. X ) -> ( ( A X. { (/) } ) u. ( ( B +c C ) X. { 1o } ) ) = ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
56	48, 55	sylan9eq 2676	. . 3 |- ( ( A e. V /\ ( B e. W /\ C e. X ) ) -> ( A +c ( B +c C ) ) = ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
57	56	3impb 1260	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A +c ( B +c C ) ) = ( ( ( A X. { (/) } ) u. ( ( B X. { (/) } ) X. { 1o } ) ) u. ( ( C X. { 1o } ) X. { 1o } ) ) )
58	45, 57	breqtrrd 4681	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A +c B ) +c C ) ~~ ( A +c ( B +c C ) ) )

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-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-1o 7560  df-er 7742  df-en 7956  df-cda 8990

This theorem is referenced by: (None)