Theorem endjusym 7294

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 7256	. . . . . . . . 9 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
2		f1of1 5582	. . . . . . . . 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 3249	. . . . . . . 8 |- A C_ _V
5		f1ores 5598	. . . . . . . 8 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ A C_ _V ) -> (inl |` A ) : A -1-1-onto-> (inl " A ) )
6	3, 4, 5	mp2an 426	. . . . . . 7 |- (inl |` A ) : A -1-1-onto-> (inl " A )
7		f1oeng 6929	. . . . . . 7 |- ( ( A e. V /\ (inl |` A ) : A -1-1-onto-> (inl " A ) ) -> A ~~ (inl " A ) )
8	6, 7	mpan2 425	. . . . . 6 |- ( A e. V -> A ~~ (inl " A ) )
9	8	ensymd 6956	. . . . 5 |- ( A e. V -> (inl " A ) ~~ A )
10		djurf1o 7257	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
11		f1of1 5582	. . . . . . . 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 5598	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ A C_ _V ) -> (inr |` A ) : A -1-1-onto-> (inr " A ) )
14	12, 4, 13	mp2an 426	. . . . . 6 |- (inr |` A ) : A -1-1-onto-> (inr " A )
15		f1oeng 6929	. . . . . 6 |- ( ( A e. V /\ (inr |` A ) : A -1-1-onto-> (inr " A ) ) -> A ~~ (inr " A ) )
16	14, 15	mpan2 425	. . . . 5 |- ( A e. V -> A ~~ (inr " A ) )
17		entr 6957	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ (inr " A ) ) -> (inl " A ) ~~ (inr " A ) )
18	9, 16, 17	syl2anc 411	. . . 4 |- ( A e. V -> (inl " A ) ~~ (inr " A ) )
19	18	adantr 276	. . 3 |- ( ( A e. V /\ B e. W ) -> (inl " A ) ~~ (inr " A ) )
20		ssv 3249	. . . . . . . 8 |- B C_ _V
21		f1ores 5598	. . . . . . . 8 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ B C_ _V ) -> (inr |` B ) : B -1-1-onto-> (inr " B ) )
22	12, 20, 21	mp2an 426	. . . . . . 7 |- (inr |` B ) : B -1-1-onto-> (inr " B )
23		f1oeng 6929	. . . . . . 7 |- ( ( B e. W /\ (inr |` B ) : B -1-1-onto-> (inr " B ) ) -> B ~~ (inr " B ) )
24	22, 23	mpan2 425	. . . . . 6 |- ( B e. W -> B ~~ (inr " B ) )
25	24	adantl 277	. . . . 5 |- ( ( A e. V /\ B e. W ) -> B ~~ (inr " B ) )
26	25	ensymd 6956	. . . 4 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ B )
27		f1ores 5598	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ B C_ _V ) -> (inl |` B ) : B -1-1-onto-> (inl " B ) )
28	3, 20, 27	mp2an 426	. . . . . 6 |- (inl |` B ) : B -1-1-onto-> (inl " B )
29		f1oeng 6929	. . . . . 6 |- ( ( B e. W /\ (inl |` B ) : B -1-1-onto-> (inl " B ) ) -> B ~~ (inl " B ) )
30	28, 29	mpan2 425	. . . . 5 |- ( B e. W -> B ~~ (inl " B ) )
31	30	adantl 277	. . . 4 |- ( ( A e. V /\ B e. W ) -> B ~~ (inl " B ) )
32		entr 6957	. . . 4 |- ( ( (inr " B ) ~~ B /\ B ~~ (inl " B ) ) -> (inr " B ) ~~ (inl " B ) )
33	26, 31, 32	syl2anc 411	. . 3 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ (inl " B ) )
34		djuin 7262	. . . 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 3399	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = ( (inr " A ) i^i (inl " B ) )
37		djuin 7262	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = (/)
38	36, 37	eqtr3i 2254	. . . 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 6990	. . 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 1274	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( (inr " A ) u. (inl " B ) ) )
42		djuun 7265	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
43		uncom 3351	. . 3 |- ( (inr " A ) u. (inl " B ) ) = ( (inl " B ) u. (inr " A ) )
44		djuun 7265	. . 3 |- ( (inl " B ) u. (inr " A ) ) = ( B ⊔ A )
45	43, 44	eqtri 2252	. 2 |- ( (inr " A ) u. (inl " B ) ) = ( B ⊔ A )
46	41, 42, 45	3brtr3g 4121	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   u. cun 3198   i^i cin 3199   C_ wss 3200   (/)c0 3494   {csn 3669   class class class wbr 4088   X. cxp 4723   |` cres 4727   "cima 4728   -1-1->wf1 5323   -1-1-onto->wf1o 5325   1oc1o 6574   ~~ cen 6906   ⊔ cdju 7235  inlcinl 7243  inrcinr 7244

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-1o 6581  df-er 6701  df-en 6909  df-dju 7236  df-inl 7245  df-inr 7246

This theorem is referenced by:  sbthom  16630