Theorem sucpw1ne3 7188

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 7187	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		1oex 6392	. . . . . 6 |- 1o e. _V
3	2	pwex 4162	. . . . 5 |- ~P 1o e. _V
4	3	sucid 4395	. . . 4 |- ~P 1o e. suc ~P 1o
5		eleq2 2230	. . . 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 2386	. 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 1343   e. wcel 2136   =/= wne 2336   ~Pcpw 3559  EXMIDwem 4173   suc csuc 4343   1oc1o 6377   3oc3o 6379

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-exmid 4174  df-iord 4344  df-on 4346  df-suc 4349  df-1o 6384  df-2o 6385  df-3o 6386

This theorem is referenced by: (None)