Theorem sucpw1ne3 7161

Description: Negated excluded middle implies that the successor of the power set of 1o is not three . (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
sucpw1ne3	|- ( -. EXMID -> suc ~P 1o =/= 3o )

Proof of Theorem sucpw1ne3

Step	Hyp	Ref	Expression
1		pw1nel3 7160	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		1oex 6368	. . . . . 6 |- 1o e. _V
3	2	pwex 4144	. . . . 5 |- ~P 1o e. _V
4	3	sucid 4377	. . . 4 |- ~P 1o e. suc ~P 1o
5		eleq2 2221	. . . 4 |- ( suc ~P 1o = 3o -> ( ~P 1o e. suc ~P 1o <-> ~P 1o e. 3o ) )
6	4, 5	mpbii 147	. . 3 |- ( suc ~P 1o = 3o -> ~P 1o e. 3o )
7	6	necon3bi 2377	. 2 |- ( -. ~P 1o e. 3o -> suc ~P 1o =/= 3o )
8	1, 7	syl 14	1 |- ( -. EXMID -> suc ~P 1o =/= 3o )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1335   e. wcel 2128   =/= wne 2327   ~Pcpw 3543  EXMIDwem 4155   suc csuc 4325   1oc1o 6353   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-tr 4063  df-exmid 4156  df-iord 4326  df-on 4328  df-suc 4331  df-1o 6360  df-2o 6361  df-3o 6362

This theorem is referenced by: (None)