Theorem dju1p1e2 7542

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 7400	. 2 |- ( (inl " 1o ) u. (inr " 1o ) ) = ( 1o ⊔ 1o )
2		djuin 7397	. . 3 |- ( (inl " 1o ) i^i (inr " 1o ) ) = (/)
3		djulf1o 7391	. . . . . . . 8 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
4		f1of1 5636	. . . . . . . 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 3270	. . . . . . 7 |- 1o C_ _V
7		f1ores 5652	. . . . . . 7 |- ( (inl : _V -1-1-> ( { (/) } X. _V ) /\ 1o C_ _V ) -> (inl |` 1o ) : 1o -1-1-onto-> (inl " 1o ) )
8	5, 6, 7	mp2an 430	. . . . . 6 |- (inl |` 1o ) : 1o -1-1-onto-> (inl " 1o )
9		1oex 6688	. . . . . . 7 |- 1o e. _V
10	9	f1oen 7038	. . . . . 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 7062	. . . 4 |- (inl " 1o ) ~~ 1o
13		djurf1o 7392	. . . . . . . 8 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
14		f1of1 5636	. . . . . . . 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 5652	. . . . . . 7 |- ( (inr : _V -1-1-> ( { 1o } X. _V ) /\ 1o C_ _V ) -> (inr |` 1o ) : 1o -1-1-onto-> (inr " 1o ) )
17	15, 6, 16	mp2an 430	. . . . . 6 |- (inr |` 1o ) : 1o -1-1-onto-> (inr " 1o )
18	9	f1oen 7038	. . . . . 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 7062	. . . 4 |- (inr " 1o ) ~~ 1o
21		pm54.43 7529	. . . 4 |- ( ( (inl " 1o ) ~~ 1o /\ (inr " 1o ) ~~ 1o ) -> ( ( (inl " 1o ) i^i (inr " 1o ) ) = (/) <-> ( (inl " 1o ) u. (inr " 1o ) ) ~~ 2o ) )
22	12, 20, 21	mp2an 430	. . 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 4151	1 |- ( 1o ⊔ 1o ) ~~ 2o

Syntax hints:   <-> wb 105   = wceq 1402   _Vcvv 2821   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3708   class class class wbr 4128   X. cxp 4770   |` cres 4774   "cima 4775   -1-1->wf1 5372   -1-1-onto->wf1o 5374   1oc1o 6673   2oc2o 6674   ~~ cen 7013   ⊔ cdju 7370  inlcinl 7378  inrcinr 7379

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 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1st 6367  df-2nd 6368  df-1o 6680  df-2o 6681  df-er 6800  df-en 7016  df-dju 7371  df-inl 7380  df-inr 7381

This theorem is referenced by:  exmidfodomrlemr  7547  exmidfodomrlemrALT  7548