Theorem djuassen 7210

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
djuassen	|- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( A ⊔ ( B ⊔ C ) ) )

Proof of Theorem djuassen

Step	Hyp	Ref	Expression
1		0ex 4127	. . . . . 6 |- (/) e. _V
2		simp1 997	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A e. V )
3		xpsnen2g 6823	. . . . . 6 |- ( ( (/) e. _V /\ A e. V ) -> ( { (/) } X. A ) ~~ A )
4	1, 2, 3	sylancr 414	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. A ) ~~ A )
5	4	ensymd 6777	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ ( { (/) } X. A ) )
6		1oex 6419	. . . . . . 7 |- 1o e. _V
7	1	snex 4182	. . . . . . . 8 |- { (/) } e. _V
8		simp2 998	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
9		xpexg 4737	. . . . . . . 8 |- ( ( { (/) } e. _V /\ B e. W ) -> ( { (/) } X. B ) e. _V )
10	7, 8, 9	sylancr 414	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) e. _V )
11		xpsnen2g 6823	. . . . . . 7 |- ( ( 1o e. _V /\ ( { (/) } X. B ) e. _V ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ ( { (/) } X. B ) )
12	6, 10, 11	sylancr 414	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ ( { (/) } X. B ) )
13		xpsnen2g 6823	. . . . . . 7 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
14	1, 8, 13	sylancr 414	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) ~~ B )
15		entr 6778	. . . . . 6 |- ( ( ( { 1o } X. ( { (/) } X. B ) ) ~~ ( { (/) } X. B ) /\ ( { (/) } X. B ) ~~ B ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ B )
16	12, 14, 15	syl2anc 411	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ B )
17	16	ensymd 6777	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { 1o } X. ( { (/) } X. B ) ) )
18		xp01disjl 6429	. . . . 5 |- ( ( { (/) } X. A ) i^i ( { 1o } X. ( { (/) } X. B ) ) ) = (/)
19	18	a1i 9	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( { (/) } X. A ) i^i ( { 1o } X. ( { (/) } X. B ) ) ) = (/) )
20		djuenun 7205	. . . 4 |- ( ( A ~~ ( { (/) } X. A ) /\ B ~~ ( { 1o } X. ( { (/) } X. B ) ) /\ ( ( { (/) } X. A ) i^i ( { 1o } X. ( { (/) } X. B ) ) ) = (/) ) -> ( A ⊔ B ) ~~ ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) )
21	5, 17, 19, 20	syl3anc 1238	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A ⊔ B ) ~~ ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) )
22	6	snex 4182	. . . . . . 7 |- { 1o } e. _V
23		simp3 999	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
24		xpexg 4737	. . . . . . 7 |- ( ( { 1o } e. _V /\ C e. X ) -> ( { 1o } X. C ) e. _V )
25	22, 23, 24	sylancr 414	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) e. _V )
26		xpsnen2g 6823	. . . . . 6 |- ( ( 1o e. _V /\ ( { 1o } X. C ) e. _V ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ ( { 1o } X. C ) )
27	6, 25, 26	sylancr 414	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ ( { 1o } X. C ) )
28		xpsnen2g 6823	. . . . . 6 |- ( ( 1o e. _V /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
29	6, 23, 28	sylancr 414	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
30		entr 6778	. . . . 5 |- ( ( ( { 1o } X. ( { 1o } X. C ) ) ~~ ( { 1o } X. C ) /\ ( { 1o } X. C ) ~~ C ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ C )
31	27, 29, 30	syl2anc 411	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ C )
32	31	ensymd 6777	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. ( { 1o } X. C ) ) )
33		indir 3384	. . . . 5 |- ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = ( ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) u. ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) )
34		xp01disjl 6429	. . . . . . 7 |- ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
35		xp01disjl 6429	. . . . . . . . 9 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
36	35	xpeq2i 4644	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( { 1o } X. (/) )
37		xpindi 4758	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) )
38		xp0 5044	. . . . . . . 8 |- ( { 1o } X. (/) ) = (/)
39	36, 37, 38	3eqtr3i 2206	. . . . . . 7 |- ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
40	34, 39	uneq12i 3287	. . . . . 6 |- ( ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) u. ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) ) = ( (/) u. (/) )
41		un0 3456	. . . . . 6 |- ( (/) u. (/) ) = (/)
42	40, 41	eqtri 2198	. . . . 5 |- ( ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) u. ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) ) = (/)
43	33, 42	eqtri 2198	. . . 4 |- ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
44	43	a1i 9	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/) )
45		djuenun 7205	. . 3 |- ( ( ( A ⊔ B ) ~~ ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) /\ C ~~ ( { 1o } X. ( { 1o } X. C ) ) /\ ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/) ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
46	21, 32, 44, 45	syl3anc 1238	. 2 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
47		df-dju 7031	. . . . . 6 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
48	47	xpeq2i 4644	. . . . 5 |- ( { 1o } X. ( B ⊔ C ) ) = ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
49		xpundi 4679	. . . . 5 |- ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
50	48, 49	eqtri 2198	. . . 4 |- ( { 1o } X. ( B ⊔ C ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
51	50	uneq2i 3286	. . 3 |- ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) ) = ( ( { (/) } X. A ) u. ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
52		df-dju 7031	. . 3 |- ( A ⊔ ( B ⊔ C ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) )
53		unass 3292	. . 3 |- ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) = ( ( { (/) } X. A ) u. ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
54	51, 52, 53	3eqtr4i 2208	. 2 |- ( A ⊔ ( B ⊔ C ) ) = ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
55	46, 54	breqtrrdi 4042	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( A ⊔ ( B ⊔ C ) ) )

Syntax hints:   -> wi 4   /\ w3a 978   = wceq 1353   e. wcel 2148   _Vcvv 2737   u. cun 3127   i^i cin 3128   (/)c0 3422   {csn 3591   class class class wbr 4000   X. cxp 4621   1oc1o 6404   ~~ cen 6732   ⊔ cdju 7030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-1st 6135  df-2nd 6136  df-1o 6411  df-er 6529  df-en 6735  df-dju 7031  df-inl 7040  df-inr 7041

This theorem is referenced by: (None)