Theorem djucomen 7276

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 6477	. . . 4 |- 1o e. _V
2		xpsnen2g 6883	. . . 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 4156	. . . 4 |- (/) e. _V
5		xpsnen2g 6883	. . . 4 |- ( ( (/) e. _V /\ B e. W ) -> ( { (/) } X. B ) ~~ B )
6	4, 5	mpan 424	. . 3 |- ( B e. W -> ( { (/) } X. B ) ~~ B )
7		ensym 6835	. . . 4 |- ( ( { 1o } X. A ) ~~ A -> A ~~ ( { 1o } X. A ) )
8		ensym 6835	. . . 4 |- ( ( { (/) } X. B ) ~~ B -> B ~~ ( { (/) } X. B ) )
9		incom 3351	. . . . . 6 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = ( ( { (/) } X. B ) i^i ( { 1o } X. A ) )
10		xp01disjl 6487	. . . . . 6 |- ( ( { (/) } X. B ) i^i ( { 1o } X. A ) ) = (/)
11	9, 10	eqtri 2214	. . . . 5 |- ( ( { 1o } X. A ) i^i ( { (/) } X. B ) ) = (/)
12		djuenun 7272	. . . . 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 7097	. . 3 |- ( B ⊔ A ) = ( ( { (/) } X. B ) u. ( { 1o } X. A ) )
17	16	equncomi 3305	. 2 |- ( B ⊔ A ) = ( ( { 1o } X. A ) u. ( { (/) } X. B ) )
18	15, 17	breqtrrdi 4071	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   u. cun 3151   i^i cin 3152   (/)c0 3446   {csn 3618   class class class wbr 4029   X. cxp 4657   1oc1o 6462   ~~ cen 6792   ⊔ cdju 7096

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-1o 6469  df-er 6587  df-en 6795  df-dju 7097  df-inl 7106  df-inr 7107

This theorem is referenced by: (None)