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Theorem sucpw1nss3 7212
Description: Negated excluded middle implies that the successor of the power set of  1o is not a subset of  3o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)
Assertion
Ref Expression
sucpw1nss3  |-  ( -. EXMID  ->  -.  suc  ~P 1o  C_  3o )

Proof of Theorem sucpw1nss3
StepHypRef Expression
1 pw1nel3 7208 . 2  |-  ( -. EXMID  ->  -.  ~P 1o  e.  3o )
2 pw1on 7203 . . 3  |-  ~P 1o  e.  On
3 sucssel 4409 . . 3  |-  ( ~P 1o  e.  On  ->  ( suc  ~P 1o  C_  3o  ->  ~P 1o  e.  3o ) )
42, 3ax-mp 5 . 2  |-  ( suc 
~P 1o  C_  3o  ->  ~P 1o  e.  3o )
51, 4nsyl 623 1  |-  ( -. EXMID  ->  -.  suc  ~P 1o  C_  3o )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 2141    C_ wss 3121   ~Pcpw 3566  EXMIDwem 4180   Oncon0 4348   suc csuc 4350   1oc1o 6388   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-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356  df-1o 6395  df-2o 6396  df-3o 6397
This theorem is referenced by:  onntri45  7218
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