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Theorem 3nelsucpw1 7169
Description: Three is not an element of the successor of the power set of  1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)
Assertion
Ref Expression
3nelsucpw1  |-  -.  3o  e.  suc  ~P 1o

Proof of Theorem 3nelsucpw1
StepHypRef Expression
1 1lt2o 6389 . . . . 5  |-  1o  e.  2o
2 elelsuc 4369 . . . . 5  |-  ( 1o  e.  2o  ->  1o  e.  suc  2o )
31, 2ax-mp 5 . . . 4  |-  1o  e.  suc  2o
4 df-3o 6365 . . . 4  |-  3o  =  suc  2o
53, 4eleqtrri 2233 . . 3  |-  1o  e.  3o
6 ssnel 4528 . . 3  |-  ( 3o  C_  1o  ->  -.  1o  e.  3o )
75, 6mt2 630 . 2  |-  -.  3o  C_  1o
8 pw1ne3 7165 . . . . . 6  |-  ~P 1o  =/=  3o
98nesymi 2373 . . . . 5  |-  -.  3o  =  ~P 1o
109a1i 9 . . . 4  |-  ( 3o  e.  suc  ~P 1o  ->  -.  3o  =  ~P 1o )
11 elsuci 4363 . . . 4  |-  ( 3o  e.  suc  ~P 1o  ->  ( 3o  e.  ~P 1o  \/  3o  =  ~P 1o ) )
1210, 11ecased 1331 . . 3  |-  ( 3o  e.  suc  ~P 1o  ->  3o  e.  ~P 1o )
1312elpwid 3554 . 2  |-  ( 3o  e.  suc  ~P 1o  ->  3o  C_  1o )
147, 13mto 652 1  |-  -.  3o  e.  suc  ~P 1o
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1335    e. wcel 2128    C_ wss 3102   ~Pcpw 3543   suc csuc 4325   1oc1o 6356   2oc2o 6357   3oc3o 6358
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 4496
This theorem depends on definitions:  df-bi 116  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-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-1o 6363  df-2o 6364  df-3o 6365
This theorem is referenced by:  onntri35  7172
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