ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dju1p1e2 Unicode version

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
StepHypRef 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 )
)
53, 4ax-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 ) )
85, 6, 7mp2an 426 . . . . . 6  |-  (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o )
9 1oex 6450 . . . . . . 7  |-  1o  e.  _V
109f1oen 6786 . . . . . 6  |-  ( (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o )  ->  1o  ~~  (inl " 1o ) )
118, 10ax-mp 5 . . . . 5  |-  1o  ~~  (inl " 1o )
1211ensymi 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 ) )
1513, 14ax-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 ) )
1715, 6, 16mp2an 426 . . . . . 6  |-  (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o )
189f1oen 6786 . . . . . 6  |-  ( (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o )  ->  1o  ~~  (inr " 1o ) )
1917, 18ax-mp 5 . . . . 5  |-  1o  ~~  (inr " 1o )
2019ensymi 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 ) )
2212, 20, 21mp2an 426 . . 3  |-  ( ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/)  <->  (
(inl " 1o )  u.  (inr " 1o ) )  ~~  2o )
232, 22mpbi 145 . 2  |-  ( (inl " 1o )  u.  (inr " 1o ) )  ~~  2o
241, 23eqbrtrri 4041 1  |-  ( 1o 1o )  ~~  2o
Colors of variables: wff set class
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
  Copyright terms: Public domain W3C validator