Theorem dju1p1e2 7307

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 7171	. 2 |- ( (inl " 1o ) u. (inr " 1o ) ) = ( 1o ⊔ 1o )
2		djuin 7168	. . 3 |- ( (inl " 1o ) i^i (inr " 1o ) ) = (/)
3		djulf1o 7162	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
4		f1of1 5523	. . . . . . . 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 3215	. . . . . . 7 |- 1o C_ _V
7		f1ores 5539	. . . . . . 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 6512	. . . . . . 7 |- 1o e. _V
10	9	f1oen 6852	. . . . . 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 6876	. . . 4 |- (inl " 1o ) ~~ 1o
13		djurf1o 7163	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
14		f1of1 5523	. . . . . . . 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 5539	. . . . . . 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 6852	. . . . . 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 6876	. . . 4 |- (inr " 1o ) ~~ 1o
21		pm54.43 7300	. . . 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 4068	1 |- ( 1o ⊔ 1o ) ~~ 2o

Syntax hints:   <-> wb 105   = wceq 1373   _Vcvv 2772   u. cun 3164   i^i cin 3165   C_ wss 3166   (/)c0 3460   {csn 3633   class class class wbr 4045   X. cxp 4674   |` cres 4678   "cima 4679   -1-1->wf1 5269   -1-1-onto->wf1o 5271   1oc1o 6497   2oc2o 6498   ~~ cen 6827   ⊔ cdju 7141  inlcinl 7149  inrcinr 7150

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-1st 6228  df-2nd 6229  df-1o 6504  df-2o 6505  df-er 6622  df-en 6830  df-dju 7142  df-inl 7151  df-inr 7152

This theorem is referenced by:  exmidfodomrlemr  7312  exmidfodomrlemrALT  7313