Theorem dju1p1e2 7227

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 7097	. 2 |- ( (inl " 1o ) u. (inr " 1o ) ) = ( 1o ⊔ 1o )
2		djuin 7094	. . 3 |- ( (inl " 1o ) i^i (inr " 1o ) ) = (/)
3		djulf1o 7088	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
4		f1of1 5479	. . . . . . . 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 3192	. . . . . . 7 |- 1o C_ _V
7		f1ores 5495	. . . . . . 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 6450	. . . . . . 7 |- 1o e. _V
10	9	f1oen 6786	. . . . . 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 6809	. . . 4 |- (inl " 1o ) ~~ 1o
13		djurf1o 7089	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
14		f1of1 5479	. . . . . . . 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 5495	. . . . . . 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 6786	. . . . . 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 6809	. . . 4 |- (inr " 1o ) ~~ 1o
21		pm54.43 7220	. . . 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 4041	1 |- ( 1o ⊔ 1o ) ~~ 2o

Syntax hints:   <-> wb 105   = wceq 1364   _Vcvv 2752   u. cun 3142   i^i cin 3143   C_ wss 3144   (/)c0 3437   {csn 3607   class class class wbr 4018   X. cxp 4642   |` cres 4646   "cima 4647   -1-1->wf1 5232   -1-1-onto->wf1o 5234   1oc1o 6435   2oc2o 6436   ~~ cen 6765   ⊔ cdju 7067  inlcinl 7075  inrcinr 7076

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-2o 6443  df-er 6560  df-en 6768  df-dju 7068  df-inl 7077  df-inr 7078

This theorem is referenced by:  exmidfodomrlemr  7232  exmidfodomrlemrALT  7233