Theorem dju1p1e2 7336

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 7195	. 2 |- ( (inl " 1o ) u. (inr " 1o ) ) = ( 1o ⊔ 1o )
2		djuin 7192	. . 3 |- ( (inl " 1o ) i^i (inr " 1o ) ) = (/)
3		djulf1o 7186	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
4		f1of1 5543	. . . . . . . 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 3223	. . . . . . 7 |- 1o C_ _V
7		f1ores 5559	. . . . . . 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 6533	. . . . . . 7 |- 1o e. _V
10	9	f1oen 6873	. . . . . 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 6897	. . . 4 |- (inl " 1o ) ~~ 1o
13		djurf1o 7187	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
14		f1of1 5543	. . . . . . . 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 5559	. . . . . . 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 6873	. . . . . 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 6897	. . . 4 |- (inr " 1o ) ~~ 1o
21		pm54.43 7324	. . . 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 4082	1 |- ( 1o ⊔ 1o ) ~~ 2o

Syntax hints:   <-> wb 105   = wceq 1373   _Vcvv 2776   u. cun 3172   i^i cin 3173   C_ wss 3174   (/)c0 3468   {csn 3643   class class class wbr 4059   X. cxp 4691   |` cres 4695   "cima 4696   -1-1->wf1 5287   -1-1-onto->wf1o 5289   1oc1o 6518   2oc2o 6519   ~~ cen 6848   ⊔ cdju 7165  inlcinl 7173  inrcinr 7174

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

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 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-dju 7166  df-inl 7175  df-inr 7176

This theorem is referenced by:  exmidfodomrlemr  7341  exmidfodomrlemrALT  7342