Theorem dju1p1e2 7199

Description: Disjoint union version of one plus one equals two. (Contributed by Jim Kingdon, 1-Jul-2022.)

Assertion

Ref	Expression
dju1p1e2	|- ( 1o ⊔ 1o ) ~~ 2o

Proof of Theorem dju1p1e2

Step	Hyp	Ref	Expression
1		djuun 7069	. 2 |- ( (inl " 1o ) u. (inr " 1o ) ) = ( 1o ⊔ 1o )
2		djuin 7066	. . 3 |- ( (inl " 1o ) i^i (inr " 1o ) ) = (/)
3		djulf1o 7060	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
4		f1of1 5462	. . . . . . . 8 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> inl : _V -1-1-> ( { (/) } X. _V ) )
5	3, 4	ax-mp 5	. . . . . . 7 |- inl : _V -1-1-> ( { (/) } X. _V )
6		ssv 3179	. . . . . . 7 |- 1o C_ _V
7		f1ores 5478	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ 1o C_ _V ) -> (inl |` 1o ) : 1o -1-1-onto-> (inl " 1o ) )
8	5, 6, 7	mp2an 426	. . . . . 6 |- (inl |` 1o ) : 1o -1-1-onto-> (inl " 1o )
9		1oex 6428	. . . . . . 7 |- 1o e. _V
10	9	f1oen 6762	. . . . . 6 |- ( (inl |` 1o ) : 1o -1-1-onto-> (inl " 1o ) -> 1o ~~ (inl " 1o ) )
11	8, 10	ax-mp 5	. . . . 5 |- 1o ~~ (inl " 1o )
12	11	ensymi 6785	. . . 4 |- (inl " 1o ) ~~ 1o
13		djurf1o 7061	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
14		f1of1 5462	. . . . . . . 8 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> inr : _V -1-1-> ( { 1o } X. _V ) )
15	13, 14	ax-mp 5	. . . . . . 7 |- inr : _V -1-1-> ( { 1o } X. _V )
16		f1ores 5478	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ 1o C_ _V ) -> (inr |` 1o ) : 1o -1-1-onto-> (inr " 1o ) )
17	15, 6, 16	mp2an 426	. . . . . 6 |- (inr |` 1o ) : 1o -1-1-onto-> (inr " 1o )
18	9	f1oen 6762	. . . . . 6 |- ( (inr |` 1o ) : 1o -1-1-onto-> (inr " 1o ) -> 1o ~~ (inr " 1o ) )
19	17, 18	ax-mp 5	. . . . 5 |- 1o ~~ (inr " 1o )
20	19	ensymi 6785	. . . 4 |- (inr " 1o ) ~~ 1o
21		pm54.43 7192	. . . 4 |- ( ( (inl " 1o ) ~~ 1o /\ (inr " 1o ) ~~ 1o ) -> ( ( (inl " 1o ) i^i (inr " 1o ) ) = (/) <-> ( (inl " 1o ) u. (inr " 1o ) ) ~~ 2o ) )
22	12, 20, 21	mp2an 426	. . 3 |- ( ( (inl " 1o ) i^i (inr " 1o ) ) = (/) <-> ( (inl " 1o ) u. (inr " 1o ) ) ~~ 2o )
23	2, 22	mpbi 145	. 2 |- ( (inl " 1o ) u. (inr " 1o ) ) ~~ 2o
24	1, 23	eqbrtrri 4028	1 |- ( 1o ⊔ 1o ) ~~ 2o

Syntax hints:   <-> wb 105   = wceq 1353   _Vcvv 2739   u. cun 3129   i^i cin 3130   C_ wss 3131   (/)c0 3424   {csn 3594   class class class wbr 4005   X. cxp 4626   |` cres 4630   "cima 4631   -1-1->wf1 5215   -1-1-onto->wf1o 5217   1oc1o 6413   2oc2o 6414   ~~ cen 6741   ⊔ cdju 7039  inlcinl 7047  inrcinr 7048

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6144  df-2nd 6145  df-1o 6420  df-2o 6421  df-er 6538  df-en 6744  df-dju 7040  df-inl 7049  df-inr 7050

This theorem is referenced by:  exmidfodomrlemr  7204  exmidfodomrlemrALT  7205