Theorem djucomen 7283

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 6482	. . . 4 |- 1o e. _V
2		xpsnen2g 6888	. . . 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 4160	. . . 4 |- (/) e. _V
5		xpsnen2g 6888	. . . 4 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	4, 5	mpan 424	. . 3 |- ( B e. W -> ( { (/) } X. B ) ~~ B )
7		ensym 6840	. . . 4 |- ( ( { 1o } X. A ) ~~ A -> A ~~ ( { 1o } X. A ) )
8		ensym 6840	. . . 4 |- ( ( { (/) } X. B ) ~~ B -> B ~~ ( { (/) } X. B ) )
9		incom 3355	. . . . . 6 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = ( ( { (/) } X. B ) i^i ( { 1o } X. A ) )
10		xp01disjl 6492	. . . . . 6 |- ( ( { (/) } X. B ) i^i ( { 1o } X. A ) ) = (/)
11	9, 10	eqtri 2217	. . . . 5 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = (/)
12		djuenun 7279	. . . . 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 1337	. . . 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 7104	. . 3 |- ( B ⊔ A ) = ( ( { (/) } X. B ) u. ( { 1o } X. A ) )
17	16	equncomi 3309	. 2 |- ( B ⊔ A ) = ( ( { 1o } X. A ) u. ( { (/) } X. B ) )
18	15, 17	breqtrrdi 4075	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   u. cun 3155   i^i cin 3156   (/)c0 3450   {csn 3622   class class class wbr 4033   X. cxp 4661   1oc1o 6467   ~~ cen 6797   ⊔ cdju 7103

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-er 6592  df-en 6800  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by: (None)