Theorem endjusym 7026

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 6988	. . . . . . . . 9 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
2		f1of1 5406	. . . . . . . . 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 3146	. . . . . . . 8 |- A C_ _V
5		f1ores 5422	. . . . . . . 8 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ A C_ _V ) -> (inl |` A ) : A -1-1-onto-> (inl " A ) )
6	3, 4, 5	mp2an 423	. . . . . . 7 |- (inl |` A ) : A -1-1-onto-> (inl " A )
7		f1oeng 6691	. . . . . . 7 |- ( ( A e. V /\ (inl |` A ) : A -1-1-onto-> (inl " A ) ) -> A ~~ (inl " A ) )
8	6, 7	mpan2 422	. . . . . 6 |- ( A e. V -> A ~~ (inl " A ) )
9	8	ensymd 6717	. . . . 5 |- ( A e. V -> (inl " A ) ~~ A )
10		djurf1o 6989	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
11		f1of1 5406	. . . . . . . 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 5422	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ A C_ _V ) -> (inr |` A ) : A -1-1-onto-> (inr " A ) )
14	12, 4, 13	mp2an 423	. . . . . 6 |- (inr |` A ) : A -1-1-onto-> (inr " A )
15		f1oeng 6691	. . . . . 6 |- ( ( A e. V /\ (inr |` A ) : A -1-1-onto-> (inr " A ) ) -> A ~~ (inr " A ) )
16	14, 15	mpan2 422	. . . . 5 |- ( A e. V -> A ~~ (inr " A ) )
17		entr 6718	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ (inr " A ) ) -> (inl " A ) ~~ (inr " A ) )
18	9, 16, 17	syl2anc 409	. . . 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 3146	. . . . . . . 8 |- B C_ _V
21		f1ores 5422	. . . . . . . 8 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ B C_ _V ) -> (inr |` B ) : B -1-1-onto-> (inr " B ) )
22	12, 20, 21	mp2an 423	. . . . . . 7 |- (inr |` B ) : B -1-1-onto-> (inr " B )
23		f1oeng 6691	. . . . . . 7 |- ( ( B e. W /\ (inr |` B ) : B -1-1-onto-> (inr " B ) ) -> B ~~ (inr " B ) )
24	22, 23	mpan2 422	. . . . . 6 |- ( B e. W -> B ~~ (inr " B ) )
25	24	adantl 275	. . . . 5 |- ( ( A e. V /\ B e. W ) -> B ~~ (inr " B ) )
26	25	ensymd 6717	. . . 4 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ B )
27		f1ores 5422	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ B C_ _V ) -> (inl |` B ) : B -1-1-onto-> (inl " B ) )
28	3, 20, 27	mp2an 423	. . . . . 6 |- (inl |` B ) : B -1-1-onto-> (inl " B )
29		f1oeng 6691	. . . . . 6 |- ( ( B e. W /\ (inl |` B ) : B -1-1-onto-> (inl " B ) ) -> B ~~ (inl " B ) )
30	28, 29	mpan2 422	. . . . 5 |- ( B e. W -> B ~~ (inl " B ) )
31	30	adantl 275	. . . 4 |- ( ( A e. V /\ B e. W ) -> B ~~ (inl " B ) )
32		entr 6718	. . . 4 |- ( ( (inr " B ) ~~ B /\ B ~~ (inl " B ) ) -> (inr " B ) ~~ (inl " B ) )
33	26, 31, 32	syl2anc 409	. . 3 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ (inl " B ) )
34		djuin 6994	. . . 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 3295	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = ( (inr " A ) i^i (inl " B ) )
37		djuin 6994	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = (/)
38	36, 37	eqtr3i 2177	. . . 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 6750	. . 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 1218	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( (inr " A ) u. (inl " B ) ) )
42		djuun 6997	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
43		uncom 3247	. . 3 |- ( (inr " A ) u. (inl " B ) ) = ( (inl " B ) u. (inr " A ) )
44		djuun 6997	. . 3 |- ( (inl " B ) u. (inr " A ) ) = ( B ⊔ A )
45	43, 44	eqtri 2175	. 2 |- ( (inr " A ) u. (inl " B ) ) = ( B ⊔ A )
46	41, 42, 45	3brtr3g 3993	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 2125   _Vcvv 2709   u. cun 3096   i^i cin 3097   C_ wss 3098   (/)c0 3390   {csn 3556   class class class wbr 3961   X. cxp 4577   |` cres 4581   "cima 4582   -1-1->wf1 5160   -1-1-onto->wf1o 5162   1oc1o 6346   ~~ cen 6672   ⊔ cdju 6967  inlcinl 6975  inrcinr 6976

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-iord 4321  df-on 4323  df-suc 4326  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-1st 6078  df-2nd 6079  df-1o 6353  df-er 6469  df-en 6675  df-dju 6968  df-inl 6977  df-inr 6978

This theorem is referenced by:  sbthom  13538