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

Theorem dju1p1e2 7161
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 7040 . 2  |-  ( (inl " 1o )  u.  (inr " 1o ) )  =  ( 1o 1o )
2 djuin 7037 . . 3  |-  ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/)
3 djulf1o 7031 . . . . . . . 8  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
4 f1of1 5439 . . . . . . . 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 3169 . . . . . . 7  |-  1o  C_  _V
7 f1ores 5455 . . . . . . 7  |-  ( (inl : _V -1-1-> ( {
(/) }  X.  _V )  /\  1o  C_  _V )  ->  (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o ) )
85, 6, 7mp2an 424 . . . . . 6  |-  (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o )
9 1oex 6400 . . . . . . 7  |-  1o  e.  _V
109f1oen 6733 . . . . . 6  |-  ( (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o )  ->  1o  ~~  (inl " 1o ) )
118, 10ax-mp 5 . . . . 5  |-  1o  ~~  (inl " 1o )
1211ensymi 6756 . . . 4  |-  (inl " 1o )  ~~  1o
13 djurf1o 7032 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
14 f1of1 5439 . . . . . . . 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 5455 . . . . . . 7  |-  ( (inr : _V -1-1-> ( { 1o }  X.  _V )  /\  1o  C_  _V )  ->  (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o ) )
1715, 6, 16mp2an 424 . . . . . 6  |-  (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o )
189f1oen 6733 . . . . . 6  |-  ( (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o )  ->  1o  ~~  (inr " 1o ) )
1917, 18ax-mp 5 . . . . 5  |-  1o  ~~  (inr " 1o )
2019ensymi 6756 . . . 4  |-  (inr " 1o )  ~~  1o
21 pm54.43 7154 . . . 4  |-  ( ( (inl " 1o ) 
~~  1o  /\  (inr " 1o )  ~~  1o )  ->  ( ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/) 
<->  ( (inl " 1o )  u.  (inr " 1o ) )  ~~  2o ) )
2212, 20, 21mp2an 424 . . 3  |-  ( ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/)  <->  (
(inl " 1o )  u.  (inr " 1o ) )  ~~  2o )
232, 22mpbi 144 . 2  |-  ( (inl " 1o )  u.  (inr " 1o ) )  ~~  2o
241, 23eqbrtrri 4010 1  |-  ( 1o 1o )  ~~  2o
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1348   _Vcvv 2730    u. cun 3119    i^i cin 3120    C_ wss 3121   (/)c0 3414   {csn 3581   class class class wbr 3987    X. cxp 4607    |` cres 4611   "cima 4612   -1-1->wf1 5193   -1-1-onto->wf1o 5195   1oc1o 6385   2oc2o 6386    ~~ cen 6712   ⊔ cdju 7010  inlcinl 7018  inrcinr 7019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-nul 4113  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6116  df-2nd 6117  df-1o 6392  df-2o 6393  df-er 6509  df-en 6715  df-dju 7011  df-inl 7020  df-inr 7021
This theorem is referenced by:  exmidfodomrlemr  7166  exmidfodomrlemrALT  7167
  Copyright terms: Public domain W3C validator