ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  3nsssucpw1 Unicode version

Theorem 3nsssucpw1 7213
Description: Negated excluded middle implies that  3o is not a subset of the successor of the power set of 
1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
Assertion
Ref Expression
3nsssucpw1  |-  ( -. EXMID  ->  -.  3o  C_  suc  ~P 1o )

Proof of Theorem 3nsssucpw1
StepHypRef Expression
1 df-3o 6397 . . . . . 6  |-  3o  =  suc  2o
21sseq1i 3173 . . . . 5  |-  ( 3o  C_  suc  ~P 1o  <->  suc  2o  C_  suc  ~P 1o )
3 1lt2o 6421 . . . . . . . . 9  |-  1o  e.  2o
4 ssnel 4553 . . . . . . . . 9  |-  ( 2o  C_  1o  ->  -.  1o  e.  2o )
53, 4mt2 635 . . . . . . . 8  |-  -.  2o  C_  1o
6 2onn 6500 . . . . . . . . . 10  |-  2o  e.  om
76elexi 2742 . . . . . . . . 9  |-  2o  e.  _V
87elpw 3572 . . . . . . . 8  |-  ( 2o  e.  ~P 1o  <->  2o  C_  1o )
95, 8mtbir 666 . . . . . . 7  |-  -.  2o  e.  ~P 1o
109a1i 9 . . . . . 6  |-  ( suc 
2o  C_  suc  ~P 1o  ->  -.  2o  e.  ~P 1o )
11 sucssel 4409 . . . . . . . . 9  |-  ( 2o  e.  om  ->  ( suc  2o  C_  suc  ~P 1o  ->  2o  e.  suc  ~P 1o ) )
126, 11ax-mp 5 . . . . . . . 8  |-  ( suc 
2o  C_  suc  ~P 1o  ->  2o  e.  suc  ~P 1o )
13 elsuci 4388 . . . . . . . 8  |-  ( 2o  e.  suc  ~P 1o  ->  ( 2o  e.  ~P 1o  \/  2o  =  ~P 1o ) )
1412, 13syl 14 . . . . . . 7  |-  ( suc 
2o  C_  suc  ~P 1o  ->  ( 2o  e.  ~P 1o  \/  2o  =  ~P 1o ) )
1514orcomd 724 . . . . . 6  |-  ( suc 
2o  C_  suc  ~P 1o  ->  ( 2o  =  ~P 1o  \/  2o  e.  ~P 1o ) )
1610, 15ecased 1344 . . . . 5  |-  ( suc 
2o  C_  suc  ~P 1o  ->  2o  =  ~P 1o )
172, 16sylbi 120 . . . 4  |-  ( 3o  C_  suc  ~P 1o  ->  2o  =  ~P 1o )
1817eqcomd 2176 . . 3  |-  ( 3o  C_  suc  ~P 1o  ->  ~P 1o  =  2o )
19 exmidpweq 6887 . . 3  |-  (EXMID  <->  ~P 1o  =  2o )
2018, 19sylibr 133 . 2  |-  ( 3o  C_  suc  ~P 1o  -> EXMID )
2120con3i 627 1  |-  ( -. EXMID  ->  -.  3o  C_  suc  ~P 1o )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 703    = wceq 1348    e. wcel 2141    C_ wss 3121   ~Pcpw 3566  EXMIDwem 4180   suc csuc 4350   omcom 4574   1oc1o 6388   2oc2o 6389   3oc3o 6390
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-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-1o 6395  df-2o 6396  df-3o 6397
This theorem is referenced by:  onntri45  7218
  Copyright terms: Public domain W3C validator