Theorem sucpw1ne3 7292

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 7291	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		1oex 6477	. . . . . 6 |- 1o e. _V
3	2	pwex 4212	. . . . 5 |- ~P 1o e. _V
4	3	sucid 4448	. . . 4 |- ~P 1o e. suc ~P 1o
5		eleq2 2257	. . . 4 |- ( suc ~P 1o = 3o -> ( ~P 1o e. suc ~P 1o <-> ~P 1o e. 3o ) )
6	4, 5	mpbii 148	. . 3 |- ( suc ~P 1o = 3o -> ~P 1o e. 3o )
7	6	necon3bi 2414	. 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 1364   e. wcel 2164   =/= wne 2364   ~Pcpw 3601  EXMIDwem 4223   suc csuc 4396   1oc1o 6462   3oc3o 6464

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-exmid 4224  df-iord 4397  df-on 4399  df-suc 4402  df-1o 6469  df-2o 6470  df-3o 6471

This theorem is referenced by: (None)