Theorem endjusym 7073

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 7035	. . . . . . . . 9 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
2		f1of1 5441	. . . . . . . . 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 3169	. . . . . . . 8 |- A C_ _V
5		f1ores 5457	. . . . . . . 8 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ A C_ _V ) -> (inl |` A ) : A -1-1-onto-> (inl " A ) )
6	3, 4, 5	mp2an 424	. . . . . . 7 |- (inl |` A ) : A -1-1-onto-> (inl " A )
7		f1oeng 6735	. . . . . . 7 |- ( ( A e. V /\ (inl |` A ) : A -1-1-onto-> (inl " A ) ) -> A ~~ (inl " A ) )
8	6, 7	mpan2 423	. . . . . 6 |- ( A e. V -> A ~~ (inl " A ) )
9	8	ensymd 6761	. . . . 5 |- ( A e. V -> (inl " A ) ~~ A )
10		djurf1o 7036	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
11		f1of1 5441	. . . . . . . 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 5457	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ A C_ _V ) -> (inr |` A ) : A -1-1-onto-> (inr " A ) )
14	12, 4, 13	mp2an 424	. . . . . 6 |- (inr |` A ) : A -1-1-onto-> (inr " A )
15		f1oeng 6735	. . . . . 6 |- ( ( A e. V /\ (inr |` A ) : A -1-1-onto-> (inr " A ) ) -> A ~~ (inr " A ) )
16	14, 15	mpan2 423	. . . . 5 |- ( A e. V -> A ~~ (inr " A ) )
17		entr 6762	. . . . 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 3169	. . . . . . . 8 |- B C_ _V
21		f1ores 5457	. . . . . . . 8 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ B C_ _V ) -> (inr |` B ) : B -1-1-onto-> (inr " B ) )
22	12, 20, 21	mp2an 424	. . . . . . 7 |- (inr |` B ) : B -1-1-onto-> (inr " B )
23		f1oeng 6735	. . . . . . 7 |- ( ( B e. W /\ (inr |` B ) : B -1-1-onto-> (inr " B ) ) -> B ~~ (inr " B ) )
24	22, 23	mpan2 423	. . . . . 6 |- ( B e. W -> B ~~ (inr " B ) )
25	24	adantl 275	. . . . 5 |- ( ( A e. V /\ B e. W ) -> B ~~ (inr " B ) )
26	25	ensymd 6761	. . . 4 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ B )
27		f1ores 5457	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ B C_ _V ) -> (inl |` B ) : B -1-1-onto-> (inl " B ) )
28	3, 20, 27	mp2an 424	. . . . . 6 |- (inl |` B ) : B -1-1-onto-> (inl " B )
29		f1oeng 6735	. . . . . 6 |- ( ( B e. W /\ (inl |` B ) : B -1-1-onto-> (inl " B ) ) -> B ~~ (inl " B ) )
30	28, 29	mpan2 423	. . . . 5 |- ( B e. W -> B ~~ (inl " B ) )
31	30	adantl 275	. . . 4 |- ( ( A e. V /\ B e. W ) -> B ~~ (inl " B ) )
32		entr 6762	. . . 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 7041	. . . 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 3319	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = ( (inr " A ) i^i (inl " B ) )
37		djuin 7041	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = (/)
38	36, 37	eqtr3i 2193	. . . 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 6794	. . 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 1234	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( (inr " A ) u. (inl " B ) ) )
42		djuun 7044	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
43		uncom 3271	. . 3 |- ( (inr " A ) u. (inl " B ) ) = ( (inl " B ) u. (inr " A ) )
44		djuun 7044	. . 3 |- ( (inl " B ) u. (inr " A ) ) = ( B ⊔ A )
45	43, 44	eqtri 2191	. 2 |- ( (inr " A ) u. (inl " B ) ) = ( B ⊔ A )
46	41, 42, 45	3brtr3g 4022	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   u. cun 3119   i^i cin 3120   C_ wss 3121   (/)c0 3414   {csn 3583   class class class wbr 3989   X. cxp 4609   |` cres 4613   "cima 4614   -1-1->wf1 5195   -1-1-onto->wf1o 5197   1oc1o 6388   ~~ cen 6716   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-er 6513  df-en 6719  df-dju 7015  df-inl 7024  df-inr 7025

This theorem is referenced by:  sbthom  14058