Theorem djuassen 7360

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 4187	. . . . . 6 |- (/) e. _V
2		simp1 1000	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A e. V )
3		xpsnen2g 6949	. . . . . 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 6898	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ ( { (/) } X. A ) )
6		1oex 6533	. . . . . . 7 |- 1o e. _V
7	1	snex 4245	. . . . . . . 8 |- { (/) } e. _V
8		simp2 1001	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
9		xpexg 4807	. . . . . . . 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 6949	. . . . . . 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 6949	. . . . . . 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 6899	. . . . . 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 6898	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { 1o } X. ( { (/) } X. B ) ) )
18		xp01disjl 6543	. . . . 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 7355	. . . 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 1250	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A ⊔ B ) ~~ ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) )
22	6	snex 4245	. . . . . . 7 |- { 1o } e. _V
23		simp3 1002	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
24		xpexg 4807	. . . . . . 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 6949	. . . . . 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 6949	. . . . . 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 6899	. . . . 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 6898	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. ( { 1o } X. C ) ) )
33		indir 3430	. . . . 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 6543	. . . . . . 7 |- ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
35		xp01disjl 6543	. . . . . . . . 9 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
36	35	xpeq2i 4714	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( { 1o } X. (/) )
37		xpindi 4831	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) )
38		xp0 5121	. . . . . . . 8 |- ( { 1o } X. (/) ) = (/)
39	36, 37, 38	3eqtr3i 2236	. . . . . . 7 |- ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
40	34, 39	uneq12i 3333	. . . . . 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 3502	. . . . . 6 |- ( (/) u. (/) ) = (/)
42	40, 41	eqtri 2228	. . . . 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 2228	. . . 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 7355	. . 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 1250	. 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 7166	. . . . . 6 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
48	47	xpeq2i 4714	. . . . 5 |- ( { 1o } X. ( B ⊔ C ) ) = ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
49		xpundi 4749	. . . . 5 |- ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
50	48, 49	eqtri 2228	. . . 4 |- ( { 1o } X. ( B ⊔ C ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
51	50	uneq2i 3332	. . 3 |- ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) ) = ( ( { (/) } X. A ) u. ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
52		df-dju 7166	. . 3 |- ( A ⊔ ( B ⊔ C ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) )
53		unass 3338	. . 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 2238	. 2 |- ( A ⊔ ( B ⊔ C ) ) = ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
55	46, 54	breqtrrdi 4101	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( A ⊔ ( B ⊔ C ) ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2178   _Vcvv 2776   u. cun 3172   i^i cin 3173   (/)c0 3468   {csn 3643   class class class wbr 4059   X. cxp 4691   1oc1o 6518   ~~ cen 6848   ⊔ cdju 7165

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-er 6643  df-en 6851  df-dju 7166  df-inl 7175  df-inr 7176

This theorem is referenced by: (None)