Theorem djuassen 7246

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 4145	. . . . . 6 |- (/) e. _V
2		simp1 999	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A e. V )
3		xpsnen2g 6855	. . . . . 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 6809	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ ( { (/) } X. A ) )
6		1oex 6449	. . . . . . 7 |- 1o e. _V
7	1	snex 4203	. . . . . . . 8 |- { (/) } e. _V
8		simp2 1000	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
9		xpexg 4758	. . . . . . . 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 6855	. . . . . . 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 6855	. . . . . . 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 6810	. . . . . 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 6809	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { 1o } X. ( { (/) } X. B ) ) )
18		xp01disjl 6459	. . . . 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 7241	. . . 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 1249	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A ⊔ B ) ~~ ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) )
22	6	snex 4203	. . . . . . 7 |- { 1o } e. _V
23		simp3 1001	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
24		xpexg 4758	. . . . . . 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 6855	. . . . . 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 6855	. . . . . 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 6810	. . . . 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 6809	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. ( { 1o } X. C ) ) )
33		indir 3399	. . . . 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 6459	. . . . . . 7 |- ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
35		xp01disjl 6459	. . . . . . . . 9 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
36	35	xpeq2i 4665	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( { 1o } X. (/) )
37		xpindi 4780	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) )
38		xp0 5066	. . . . . . . 8 |- ( { 1o } X. (/) ) = (/)
39	36, 37, 38	3eqtr3i 2218	. . . . . . 7 |- ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
40	34, 39	uneq12i 3302	. . . . . 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 3471	. . . . . 6 |- ( (/) u. (/) ) = (/)
42	40, 41	eqtri 2210	. . . . 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 2210	. . . 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 7241	. . 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 1249	. 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 7067	. . . . . 6 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
48	47	xpeq2i 4665	. . . . 5 |- ( { 1o } X. ( B ⊔ C ) ) = ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
49		xpundi 4700	. . . . 5 |- ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
50	48, 49	eqtri 2210	. . . 4 |- ( { 1o } X. ( B ⊔ C ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
51	50	uneq2i 3301	. . 3 |- ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) ) = ( ( { (/) } X. A ) u. ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
52		df-dju 7067	. . 3 |- ( A ⊔ ( B ⊔ C ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) )
53		unass 3307	. . 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 2220	. 2 |- ( A ⊔ ( B ⊔ C ) ) = ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
55	46, 54	breqtrrdi 4060	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( A ⊔ ( B ⊔ C ) ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   _Vcvv 2752   u. cun 3142   i^i cin 3143   (/)c0 3437   {csn 3607   class class class wbr 4018   X. cxp 4642   1oc1o 6434   ~~ cen 6764   ⊔ cdju 7066

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6165  df-2nd 6166  df-1o 6441  df-er 6559  df-en 6767  df-dju 7067  df-inl 7076  df-inr 7077

This theorem is referenced by: (None)