Theorem endjusym 6981

Description: Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)

Assertion

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

Proof of Theorem endjusym

Step	Hyp	Ref	Expression
1		djulf1o 6943	. . . . . . . . 9 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
2		f1of1 5366	. . . . . . . . 9 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> inl : _V -1-1-> ( { (/) } X. _V ) )
3	1, 2	ax-mp 5	. . . . . . . 8 |- inl : _V -1-1-> ( { (/) } X. _V )
4		ssv 3119	. . . . . . . 8 |- A C_ _V
5		f1ores 5382	. . . . . . . 8 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ A C_ _V ) -> (inl |` A ) : A -1-1-onto-> (inl " A ) )
6	3, 4, 5	mp2an 422	. . . . . . 7 |- (inl |` A ) : A -1-1-onto-> (inl " A )
7		f1oeng 6651	. . . . . . 7 |- ( ( A e. V /\ (inl |` A ) : A -1-1-onto-> (inl " A ) ) -> A ~~ (inl " A ) )
8	6, 7	mpan2 421	. . . . . 6 |- ( A e. V -> A ~~ (inl " A ) )
9	8	ensymd 6677	. . . . 5 |- ( A e. V -> (inl " A ) ~~ A )
10		djurf1o 6944	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
11		f1of1 5366	. . . . . . . 8 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> inr : _V -1-1-> ( { 1o } X. _V ) )
12	10, 11	ax-mp 5	. . . . . . 7 |- inr : _V -1-1-> ( { 1o } X. _V )
13		f1ores 5382	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ A C_ _V ) -> (inr |` A ) : A -1-1-onto-> (inr " A ) )
14	12, 4, 13	mp2an 422	. . . . . 6 |- (inr |` A ) : A -1-1-onto-> (inr " A )
15		f1oeng 6651	. . . . . 6 |- ( ( A e. V /\ (inr |` A ) : A -1-1-onto-> (inr " A ) ) -> A ~~ (inr " A ) )
16	14, 15	mpan2 421	. . . . 5 |- ( A e. V -> A ~~ (inr " A ) )
17		entr 6678	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ (inr " A ) ) -> (inl " A ) ~~ (inr " A ) )
18	9, 16, 17	syl2anc 408	. . . 4 |- ( A e. V -> (inl " A ) ~~ (inr " A ) )
19	18	adantr 274	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl " A ) ~~ (inr " A ) )
20		ssv 3119	. . . . . . . 8 |- B C_ _V
21		f1ores 5382	. . . . . . . 8 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ B C_ _V ) -> (inr |` B ) : B -1-1-onto-> (inr " B ) )
22	12, 20, 21	mp2an 422	. . . . . . 7 |- (inr |` B ) : B -1-1-onto-> (inr " B )
23		f1oeng 6651	. . . . . . 7 |- ( ( B e. W /\ (inr |` B ) : B -1-1-onto-> (inr " B ) ) -> B ~~ (inr " B ) )
24	22, 23	mpan2 421	. . . . . 6 |- ( B e. W -> B ~~ (inr " B ) )
25	24	adantl 275	. . . . 5 |- ( ( A e. V /\ B e. W ) -> B ~~ (inr " B ) )
26	25	ensymd 6677	. . . 4 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ B )
27		f1ores 5382	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ B C_ _V ) -> (inl |` B ) : B -1-1-onto-> (inl " B ) )
28	3, 20, 27	mp2an 422	. . . . . 6 |- (inl |` B ) : B -1-1-onto-> (inl " B )
29		f1oeng 6651	. . . . . 6 |- ( ( B e. W /\ (inl |` B ) : B -1-1-onto-> (inl " B ) ) -> B ~~ (inl " B ) )
30	28, 29	mpan2 421	. . . . 5 |- ( B e. W -> B ~~ (inl " B ) )
31	30	adantl 275	. . . 4 |- ( ( A e. V /\ B e. W ) -> B ~~ (inl " B ) )
32		entr 6678	. . . 4 |- ( ( (inr " B ) ~~ B /\ B ~~ (inl " B ) ) -> (inr " B ) ~~ (inl " B ) )
33	26, 31, 32	syl2anc 408	. . 3 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ (inl " B ) )
34		djuin 6949	. . . 4 |- ( (inl " A ) i^i (inr " B ) ) = (/)
35	34	a1i 9	. . 3 |- ( ( A e. V /\ B e. W ) -> ( (inl " A ) i^i (inr " B ) ) = (/) )
36		incom 3268	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = ( (inr " A ) i^i (inl " B ) )
37		djuin 6949	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = (/)
38	36, 37	eqtr3i 2162	. . . 4 |- ( (inr " A ) i^i (inl " B ) ) = (/)
39	38	a1i 9	. . 3 |- ( ( A e. V /\ B e. W ) -> ( (inr " A ) i^i (inl " B ) ) = (/) )
40		unen 6710	. . 3 |- ( ( ( (inl " A ) ~~ (inr " A ) /\ (inr " B ) ~~ (inl " B ) ) /\ ( ( (inl " A ) i^i (inr " B ) ) = (/) /\ ( (inr " A ) i^i (inl " B ) ) = (/) ) ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( (inr " A ) u. (inl " B ) ) )
41	19, 33, 35, 39, 40	syl22anc 1217	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( (inr " A ) u. (inl " B ) ) )
42		djuun 6952	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
43		uncom 3220	. . 3 |- ( (inr " A ) u. (inl " B ) ) = ( (inl " B ) u. (inr " A ) )
44		djuun 6952	. . 3 |- ( (inl " B ) u. (inr " A ) ) = ( B ⊔ A )
45	43, 44	eqtri 2160	. 2 |- ( (inr " A ) u. (inl " B ) ) = ( B ⊔ A )
46	41, 42, 45	3brtr3g 3961	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   u. cun 3069   i^i cin 3070   C_ wss 3071   (/)c0 3363   {csn 3527   class class class wbr 3929   X. cxp 4537   |` cres 4541   "cima 4542   -1-1->wf1 5120   -1-1-onto->wf1o 5122   1oc1o 6306   ~~ cen 6632   ⊔ cdju 6922  inlcinl 6930  inrcinr 6931

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-1o 6313  df-er 6429  df-en 6635  df-dju 6923  df-inl 6932  df-inr 6933

This theorem is referenced by:  sbthom  13221