Theorem djuassen 7563

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 4255	. . . . . 6 |- (/) e. _V
2		simp1 1028	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A e. V )
3		xpsnen2g 7117	. . . . . 6 |- ( ( (/) e. _V /\ A e. V ) -> ( { (/) } X. A ) ~~ A )
4	1, 2, 3	sylancr 418	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. A ) ~~ A )
5	4	ensymd 7060	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> A ~~ ( { (/) } X. A ) )
6		1oex 6685	. . . . . . 7 |- 1o e. _V
7	1	snex 4317	. . . . . . . 8 |- { (/) } e. _V
8		simp2 1029	. . . . . . . 8 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B e. W )
9		xpexg 4884	. . . . . . . 8 |- ( ( { (/) } e. _V /\ B e. W ) -> ( { (/) } X. B ) e. _V )
10	7, 8, 9	sylancr 418	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) e. _V )
11		xpsnen2g 7117	. . . . . . 7 |- ( ( 1o e. _V /\ ( { (/) } X. B ) e. _V ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ ( { (/) } X. B ) )
12	6, 10, 11	sylancr 418	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ ( { (/) } X. B ) )
13		xpsnen2g 7117	. . . . . . 7 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
14	1, 8, 13	sylancr 418	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { (/) } X. B ) ~~ B )
15		entr 7061	. . . . . 6 |- ( ( ( { 1o } X. ( { (/) } X. B ) ) ~~ ( { (/) } X. B ) /\ ( { (/) } X. B ) ~~ B ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ B )
16	12, 14, 15	syl2anc 415	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { (/) } X. B ) ) ~~ B )
17	16	ensymd 7060	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> B ~~ ( { 1o } X. ( { (/) } X. B ) ) )
18		xp01disjl 6697	. . . . 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 7558	. . . 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 1278	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( A ⊔ B ) ~~ ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) )
22	6	snex 4317	. . . . . . 7 |- { 1o } e. _V
23		simp3 1030	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C e. X )
24		xpexg 4884	. . . . . . 7 |- ( ( { 1o } e. _V /\ C e. X ) -> ( { 1o } X. C ) e. _V )
25	22, 23, 24	sylancr 418	. . . . . 6 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) e. _V )
26		xpsnen2g 7117	. . . . . 6 |- ( ( 1o e. _V /\ ( { 1o } X. C ) e. _V ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ ( { 1o } X. C ) )
27	6, 25, 26	sylancr 418	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ ( { 1o } X. C ) )
28		xpsnen2g 7117	. . . . . 6 |- ( ( 1o e. _V /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
29	6, 23, 28	sylancr 418	. . . . 5 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. C ) ~~ C )
30		entr 7061	. . . . 5 |- ( ( ( { 1o } X. ( { 1o } X. C ) ) ~~ ( { 1o } X. C ) /\ ( { 1o } X. C ) ~~ C ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ C )
31	27, 29, 30	syl2anc 415	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( { 1o } X. ( { 1o } X. C ) ) ~~ C )
32	31	ensymd 7060	. . 3 |- ( ( A e. V /\ B e. W /\ C e. X ) -> C ~~ ( { 1o } X. ( { 1o } X. C ) ) )
33		indir 3480	. . . . 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 6697	. . . . . . 7 |- ( ( { (/) } X. A ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
35		xp01disjl 6697	. . . . . . . . 9 |- ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) = (/)
36	35	xpeq2i 4790	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( { 1o } X. (/) )
37		xpindi 4910	. . . . . . . 8 |- ( { 1o } X. ( ( { (/) } X. B ) i^i ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) )
38		xp0 5202	. . . . . . . 8 |- ( { 1o } X. (/) ) = (/)
39	36, 37, 38	3eqtr3i 2267	. . . . . . 7 |- ( ( { 1o } X. ( { (/) } X. B ) ) i^i ( { 1o } X. ( { 1o } X. C ) ) ) = (/)
40	34, 39	uneq12i 3381	. . . . . 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 3556	. . . . . 6 |- ( (/) u. (/) ) = (/)
42	40, 41	eqtri 2259	. . . . 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 2259	. . . 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 7558	. . 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 1278	. 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 7368	. . . . . 6 |- ( B ⊔ C ) = ( ( { (/) } X. B ) u. ( { 1o } X. C ) )
48	47	xpeq2i 4790	. . . . 5 |- ( { 1o } X. ( B ⊔ C ) ) = ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) )
49		xpundi 4826	. . . . 5 |- ( { 1o } X. ( ( { (/) } X. B ) u. ( { 1o } X. C ) ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
50	48, 49	eqtri 2259	. . . 4 |- ( { 1o } X. ( B ⊔ C ) ) = ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
51	50	uneq2i 3380	. . 3 |- ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) ) = ( ( { (/) } X. A ) u. ( ( { 1o } X. ( { (/) } X. B ) ) u. ( { 1o } X. ( { 1o } X. C ) ) ) )
52		df-dju 7368	. . 3 |- ( A ⊔ ( B ⊔ C ) ) = ( ( { (/) } X. A ) u. ( { 1o } X. ( B ⊔ C ) ) )
53		unass 3386	. . 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 2269	. 2 |- ( A ⊔ ( B ⊔ C ) ) = ( ( ( { (/) } X. A ) u. ( { 1o } X. ( { (/) } X. B ) ) ) u. ( { 1o } X. ( { 1o } X. C ) ) )
55	46, 54	breqtrrdi 4167	1 |- ( ( A e. V /\ B e. W /\ C e. X ) -> ( ( A ⊔ B ) ⊔ C ) ~~ ( A ⊔ ( B ⊔ C ) ) )

Syntax hints:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3705   class class class wbr 4125   X. cxp 4767   1oc1o 6670   ~~ cen 7010   ⊔ cdju 7367

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-er 6797  df-en 7013  df-dju 7368  df-inl 7377  df-inr 7378

This theorem is referenced by: (None)