Theorem endjusym 7426

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 7388	. . . . . . . . 9 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
2		f1of1 5633	. . . . . . . . 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 3270	. . . . . . . 8 |- A C_ _V
5		f1ores 5649	. . . . . . . 8 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ A C_ _V ) -> (inl |` A ) : A -1-1-onto-> (inl " A ) )
6	3, 4, 5	mp2an 430	. . . . . . 7 |- (inl |` A ) : A -1-1-onto-> (inl " A )
7		f1oeng 7033	. . . . . . 7 |- ( ( A e. V /\ (inl |` A ) : A -1-1-onto-> (inl " A ) ) -> A ~~ (inl " A ) )
8	6, 7	mpan2 429	. . . . . 6 |- ( A e. V -> A ~~ (inl " A ) )
9	8	ensymd 7060	. . . . 5 |- ( A e. V -> (inl " A ) ~~ A )
10		djurf1o 7389	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
11		f1of1 5633	. . . . . . . 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 5649	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ A C_ _V ) -> (inr |` A ) : A -1-1-onto-> (inr " A ) )
14	12, 4, 13	mp2an 430	. . . . . 6 |- (inr |` A ) : A -1-1-onto-> (inr " A )
15		f1oeng 7033	. . . . . 6 |- ( ( A e. V /\ (inr |` A ) : A -1-1-onto-> (inr " A ) ) -> A ~~ (inr " A ) )
16	14, 15	mpan2 429	. . . . 5 |- ( A e. V -> A ~~ (inr " A ) )
17		entr 7061	. . . . 5 |- ( ( (inl " A ) ~~ A /\ A ~~ (inr " A ) ) -> (inl " A ) ~~ (inr " A ) )
18	9, 16, 17	syl2anc 415	. . . 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 3270	. . . . . . . 8 |- B C_ _V
21		f1ores 5649	. . . . . . . 8 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ B C_ _V ) -> (inr |` B ) : B -1-1-onto-> (inr " B ) )
22	12, 20, 21	mp2an 430	. . . . . . 7 |- (inr |` B ) : B -1-1-onto-> (inr " B )
23		f1oeng 7033	. . . . . . 7 |- ( ( B e. W /\ (inr |` B ) : B -1-1-onto-> (inr " B ) ) -> B ~~ (inr " B ) )
24	22, 23	mpan2 429	. . . . . 6 |- ( B e. W -> B ~~ (inr " B ) )
25	24	adantl 277	. . . . 5 |- ( ( A e. V /\ B e. W ) -> B ~~ (inr " B ) )
26	25	ensymd 7060	. . . 4 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ B )
27		f1ores 5649	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ B C_ _V ) -> (inl |` B ) : B -1-1-onto-> (inl " B ) )
28	3, 20, 27	mp2an 430	. . . . . 6 |- (inl |` B ) : B -1-1-onto-> (inl " B )
29		f1oeng 7033	. . . . . 6 |- ( ( B e. W /\ (inl |` B ) : B -1-1-onto-> (inl " B ) ) -> B ~~ (inl " B ) )
30	28, 29	mpan2 429	. . . . 5 |- ( B e. W -> B ~~ (inl " B ) )
31	30	adantl 277	. . . 4 |- ( ( A e. V /\ B e. W ) -> B ~~ (inl " B ) )
32		entr 7061	. . . 4 |- ( ( (inr " B ) ~~ B /\ B ~~ (inl " B ) ) -> (inr " B ) ~~ (inl " B ) )
33	26, 31, 32	syl2anc 415	. . 3 |- ( ( A e. V /\ B e. W ) -> (inr " B ) ~~ (inl " B ) )
34		djuin 7394	. . . 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 3421	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = ( (inr " A ) i^i (inl " B ) )
37		djuin 7394	. . . . 5 |- ( (inl " B ) i^i (inr " A ) ) = (/)
38	36, 37	eqtr3i 2261	. . . 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 7095	. . 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 1279	. 2 |- ( ( A e. V /\ B e. W ) -> ( (inl " A ) u. (inr " B ) ) ~~ ( (inr " A ) u. (inl " B ) ) )
42		djuun 7397	. 2 |- ( (inl " A ) u. (inr " B ) ) = ( A ⊔ B )
43		uncom 3373	. . 3 |- ( (inr " A ) u. (inl " B ) ) = ( (inl " B ) u. (inr " A ) )
44		djuun 7397	. . 3 |- ( (inl " B ) u. (inr " A ) ) = ( B ⊔ A )
45	43, 44	eqtri 2259	. 2 |- ( (inr " A ) u. (inl " B ) ) = ( B ⊔ A )
46	41, 42, 45	3brtr3g 4158	1 |- ( ( A e. V /\ B e. W ) -> ( A ⊔ B ) ~~ ( B ⊔ A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3705   class class class wbr 4125   X. cxp 4767   |` cres 4771   "cima 4772   -1-1->wf1 5369   -1-1-onto->wf1o 5371   1oc1o 6670   ~~ cen 7010   ⊔ cdju 7367  inlcinl 7375  inrcinr 7376

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1st 6364  df-2nd 6365  df-1o 6677  df-er 6797  df-en 7013  df-dju 7368  df-inl 7377  df-inr 7378

This theorem is referenced by:  sbthom  16976