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Theorem dju1p1e2 6823
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 6760 . 2  |-  ( (inl " 1o )  u.  (inr " 1o ) )  =  ( 1o 1o )
2 djuin 6756 . . 3  |-  ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/)
3 djulf1o 6750 . . . . . . . 8  |- inl : _V -1-1-onto-> ( { (/) }  X.  _V )
4 f1of1 5252 . . . . . . . 8  |-  (inl : _V
-1-1-onto-> ( { (/) }  X.  _V )  -> inl : _V -1-1-> ( {
(/) }  X.  _V )
)
53, 4ax-mp 7 . . . . . . 7  |- inl : _V -1-1-> ( { (/) }  X.  _V )
6 ssv 3046 . . . . . . 7  |-  1o  C_  _V
7 f1ores 5268 . . . . . . 7  |-  ( (inl : _V -1-1-> ( {
(/) }  X.  _V )  /\  1o  C_  _V )  ->  (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o ) )
85, 6, 7mp2an 417 . . . . . 6  |-  (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o )
9 1oex 6189 . . . . . . 7  |-  1o  e.  _V
109f1oen 6476 . . . . . 6  |-  ( (inl  |`  1o ) : 1o -1-1-onto-> (inl " 1o )  ->  1o  ~~  (inl " 1o ) )
118, 10ax-mp 7 . . . . 5  |-  1o  ~~  (inl " 1o )
1211ensymi 6499 . . . 4  |-  (inl " 1o )  ~~  1o
13 djurf1o 6751 . . . . . . . 8  |- inr : _V -1-1-onto-> ( { 1o }  X.  _V )
14 f1of1 5252 . . . . . . . 8  |-  (inr : _V
-1-1-onto-> ( { 1o }  X.  _V )  -> inr : _V -1-1-> ( { 1o }  X.  _V ) )
1513, 14ax-mp 7 . . . . . . 7  |- inr : _V -1-1-> ( { 1o }  X.  _V )
16 f1ores 5268 . . . . . . 7  |-  ( (inr : _V -1-1-> ( { 1o }  X.  _V )  /\  1o  C_  _V )  ->  (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o ) )
1715, 6, 16mp2an 417 . . . . . 6  |-  (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o )
189f1oen 6476 . . . . . 6  |-  ( (inr  |`  1o ) : 1o -1-1-onto-> (inr " 1o )  ->  1o  ~~  (inr " 1o ) )
1917, 18ax-mp 7 . . . . 5  |-  1o  ~~  (inr " 1o )
2019ensymi 6499 . . . 4  |-  (inr " 1o )  ~~  1o
21 pm54.43 6818 . . . 4  |-  ( ( (inl " 1o ) 
~~  1o  /\  (inr " 1o )  ~~  1o )  ->  ( ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/) 
<->  ( (inl " 1o )  u.  (inr " 1o ) )  ~~  2o ) )
2212, 20, 21mp2an 417 . . 3  |-  ( ( (inl " 1o )  i^i  (inr " 1o ) )  =  (/)  <->  (
(inl " 1o )  u.  (inr " 1o ) )  ~~  2o )
232, 22mpbi 143 . 2  |-  ( (inl " 1o )  u.  (inr " 1o ) )  ~~  2o
241, 23eqbrtrri 3866 1  |-  ( 1o 1o )  ~~  2o
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289   _Vcvv 2619    u. cun 2997    i^i cin 2998    C_ wss 2999   (/)c0 3286   {csn 3446   class class class wbr 3845    X. cxp 4436    |` cres 4440   "cima 4441   -1-1->wf1 5012   -1-1-onto->wf1o 5014   1oc1o 6174   2oc2o 6175    ~~ cen 6455   ⊔ cdju 6730  inlcinl 6737  inrcinr 6738
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-2o 6182  df-er 6292  df-en 6458  df-dju 6731  df-inl 6739  df-inr 6740
This theorem is referenced by:  exmidfodomrlemr  6828  exmidfodomrlemrALT  6829
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