Theorem sucpw1ne3 7542

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 7541	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		1oex 6655	. . . . . 6 |- 1o e. _V
3	2	pwex 4296	. . . . 5 |- ~P 1o e. _V
4	3	sucid 4538	. . . 4 |- ~P 1o e. suc ~P 1o
5		eleq2 2296	. . . 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 2462	. 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 1398   e. wcel 2203   =/= wne 2412   ~Pcpw 3669  EXMIDwem 4307   suc csuc 4486   1oc1o 6640   3oc3o 6642

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-tr 4209  df-exmid 4308  df-iord 4487  df-on 4489  df-suc 4492  df-1o 6647  df-2o 6648  df-3o 6649

This theorem is referenced by: (None)