Theorem djucomen 7217

Description: Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258. (Contributed by NM, 24-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
djucomen	|- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Proof of Theorem djucomen

Step	Hyp	Ref	Expression
1		1oex 6427	. . . 4 |- 1o e. _V
2		xpsnen2g 6831	. . . 4 |- ( ( 1o e. _V /\ A e. V ) -> ( { 1o } X. A ) ~~ A )
3	1, 2	mpan 424	. . 3 |- ( A e. V -> ( { 1o } X. A ) ~~ A )
4		0ex 4132	. . . 4 |- (/) e. _V
5		xpsnen2g 6831	. . . 4 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	4, 5	mpan 424	. . 3 |- ( B e. W -> ( { (/) } X. B ) ~~ B )
7		ensym 6783	. . . 4 |- ( ( { 1o } X. A ) ~~ A -> A ~~ ( { 1o } X. A ) )
8		ensym 6783	. . . 4 |- ( ( { (/) } X. B ) ~~ B -> B ~~ ( { (/) } X. B ) )
9		incom 3329	. . . . . 6 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = ( ( { (/) } X. B ) i^i ( { 1o } X. A ) )
10		xp01disjl 6437	. . . . . 6 |- ( ( { (/) } X. B ) i^i ( { 1o } X. A ) ) = (/)
11	9, 10	eqtri 2198	. . . . 5 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = (/)
12		djuenun 7213	. . . . 5 |- ( ( A ~~ ( { 1o } X. A ) /\ B ~~ ( { (/) } X. B ) /\ ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = (/) ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
13	11, 12	mp3an3 1326	. . . 4 |- ( ( A ~~ ( { 1o } X. A ) /\ B ~~ ( { (/) } X. B ) ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
14	7, 8, 13	syl2an 289	. . 3 |- ( ( ( { 1o } X. A ) ~~ A /\ ( { (/) } X. B ) ~~ B ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
15	3, 6, 14	syl2an 289	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
16		df-dju 7039	. . 3 |- ( B ⊔ A ) = ( ( { (/) } X. B ) u. ( { 1o } X. A ) )
17	16	equncomi 3283	. 2 |- ( B ⊔ A ) = ( ( { 1o } X. A ) u. ( { (/) } X. B ) )
18	15, 17	breqtrrdi 4047	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   u. cun 3129   i^i cin 3130   (/)c0 3424   {csn 3594   class class class wbr 4005   X. cxp 4626   1oc1o 6412   ~~ cen 6740   ⊔ cdju 7038

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-er 6537  df-en 6743  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by: (None)