Theorem sucpw1ne3 7555

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 7554	. 2 |- ( -. EXMID -> -. ~P 1o e. 3o )
2		1oex 6668	. . . . . 6 |- 1o e. _V
3	2	pwex 4301	. . . . 5 |- ~P 1o e. _V
4	3	sucid 4543	. . . 4 |- ~P 1o e. suc ~P 1o
5		eleq2 2298	. . . 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 2464	. 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 2205   =/= wne 2414   ~Pcpw 3674  EXMIDwem 4312   suc csuc 4491   1oc1o 6653   3oc3o 6655

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-exmid 4313  df-iord 4492  df-on 4494  df-suc 4497  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by: (None)