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Theorem 3nsssucpw1 7166
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 6362 . . . . . 6  |-  3o  =  suc  2o
21sseq1i 3154 . . . . 5  |-  ( 3o  C_  suc  ~P 1o  <->  suc  2o  C_  suc  ~P 1o )
3 1lt2o 6386 . . . . . . . . 9  |-  1o  e.  2o
4 ssnel 4527 . . . . . . . . 9  |-  ( 2o  C_  1o  ->  -.  1o  e.  2o )
53, 4mt2 630 . . . . . . . 8  |-  -.  2o  C_  1o
6 2onn 6465 . . . . . . . . . 10  |-  2o  e.  om
76elexi 2724 . . . . . . . . 9  |-  2o  e.  _V
87elpw 3549 . . . . . . . 8  |-  ( 2o  e.  ~P 1o  <->  2o  C_  1o )
95, 8mtbir 661 . . . . . . 7  |-  -.  2o  e.  ~P 1o
109a1i 9 . . . . . 6  |-  ( suc 
2o  C_  suc  ~P 1o  ->  -.  2o  e.  ~P 1o )
11 sucssel 4384 . . . . . . . . 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 4363 . . . . . . . 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 719 . . . . . 6  |-  ( suc 
2o  C_  suc  ~P 1o  ->  ( 2o  =  ~P 1o  \/  2o  e.  ~P 1o ) )
1610, 15ecased 1331 . . . . 5  |-  ( suc 
2o  C_  suc  ~P 1o  ->  2o  =  ~P 1o )
172, 16sylbi 120 . . . 4  |-  ( 3o  C_  suc  ~P 1o  ->  2o  =  ~P 1o )
1817eqcomd 2163 . . 3  |-  ( 3o  C_  suc  ~P 1o  ->  ~P 1o  =  2o )
19 exmidpweq 6851 . . 3  |-  (EXMID  <->  ~P 1o  =  2o )
2018, 19sylibr 133 . 2  |-  ( 3o  C_  suc  ~P 1o  -> EXMID )
2120con3i 622 1  |-  ( -. EXMID  ->  -.  3o  C_  suc  ~P 1o )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 698    = wceq 1335    e. wcel 2128    C_ wss 3102   ~Pcpw 3543  EXMIDwem 4155   suc csuc 4325   omcom 4548   1oc1o 6353   2oc2o 6354   3oc3o 6355
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-exmid 4156  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-1o 6360  df-2o 6361  df-3o 6362
This theorem is referenced by:  onntri45  7171
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