Theorem djucomen 7172

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 6392	. . . 4 |- 1o e. _V
2		xpsnen2g 6795	. . . 4 |- ( ( 1o e. _V /\ A e. V ) -> ( { 1o } X. A ) ~~ A )
3	1, 2	mpan 421	. . 3 |- ( A e. V -> ( { 1o } X. A ) ~~ A )
4		0ex 4109	. . . 4 |- (/) e. _V
5		xpsnen2g 6795	. . . 4 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	4, 5	mpan 421	. . 3 |- ( B e. W -> ( { (/) } X. B ) ~~ B )
7		ensym 6747	. . . 4 |- ( ( { 1o } X. A ) ~~ A -> A ~~ ( { 1o } X. A ) )
8		ensym 6747	. . . 4 |- ( ( { (/) } X. B ) ~~ B -> B ~~ ( { (/) } X. B ) )
9		incom 3314	. . . . . 6 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = ( ( { (/) } X. B ) i^i ( { 1o } X. A ) )
10		xp01disjl 6402	. . . . . 6 |- ( ( { (/) } X. B ) i^i ( { 1o } X. A ) ) = (/)
11	9, 10	eqtri 2186	. . . . 5 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = (/)
12		djuenun 7168	. . . . 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 1316	. . . 4 |- ( ( A ~~ ( { 1o } X. A ) /\ B ~~ ( { (/) } X. B ) ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
14	7, 8, 13	syl2an 287	. . 3 |- ( ( ( { 1o } X. A ) ~~ A /\ ( { (/) } X. B ) ~~ B ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
15	3, 6, 14	syl2an 287	. 2 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( ( { 1o } X. A ) u. ( { (/) } X. B ) ) )
16		df-dju 7003	. . 3 |- ( B ⊔ A ) = ( ( { (/) } X. B ) u. ( { 1o } X. A ) )
17	16	equncomi 3268	. 2 |- ( B ⊔ A ) = ( ( { 1o } X. A ) u. ( { (/) } X. B ) )
18	15, 17	breqtrrdi 4024	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   u. cun 3114   i^i cin 3115   (/)c0 3409   {csn 3576   class class class wbr 3982   X. cxp 4602   1oc1o 6377   ~~ cen 6704   ⊔ cdju 7002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-1o 6384  df-er 6501  df-en 6707  df-dju 7003  df-inl 7012  df-inr 7013

This theorem is referenced by: (None)