Theorem 3nsssucpw1 7548

Description: Negated excluded middle implies that 3o is not a subset of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 31-Jul-2024.)

Assertion

Ref	Expression
3nsssucpw1	|- ( -. EXMID -> -. 3o C_ suc ~P 1o )

Proof of Theorem 3nsssucpw1

Step	Hyp	Ref	Expression
1		df-3o 6651	. . . . . 6 |- 3o = suc 2o
2	1	sseq1i 3266	. . . . 5 |- ( 3o C_ suc ~P 1o <-> suc 2o C_ suc ~P 1o )
3		1lt2o 6677	. . . . . . . . 9 |- 1o e. 2o
4		ssnel 4693	. . . . . . . . 9 |- ( 2o C_ 1o -> -. 1o e. 2o )
5	3, 4	mt2 645	. . . . . . . 8 |- -. 2o C_ 1o
6		2onn 6756	. . . . . . . . . 10 |- 2o e. om
7	6	elexi 2828	. . . . . . . . 9 |- 2o e. _V
8	7	elpw 3677	. . . . . . . 8 |- ( 2o e. ~P 1o <-> 2o C_ 1o )
9	5, 8	mtbir 678	. . . . . . 7 |- -. 2o e. ~P 1o
10	9	a1i 9	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> -. 2o e. ~P 1o )
11		sucssel 4547	. . . . . . . . 9 |- ( 2o e. om -> ( suc 2o C_ suc ~P 1o -> 2o e. suc ~P 1o ) )
12	6, 11	ax-mp 5	. . . . . . . 8 |- ( suc 2o C_ suc ~P 1o -> 2o e. suc ~P 1o )
13		elsuci 4526	. . . . . . . 8 |- ( 2o e. suc ~P 1o -> ( 2o e. ~P 1o \/ 2o = ~P 1o ) )
14	12, 13	syl 14	. . . . . . 7 |- ( suc 2o C_ suc ~P 1o -> ( 2o e. ~P 1o \/ 2o = ~P 1o ) )
15	14	orcomd 737	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> ( 2o = ~P 1o \/ 2o e. ~P 1o ) )
16	10, 15	ecased 1386	. . . . 5 |- ( suc 2o C_ suc ~P 1o -> 2o = ~P 1o )
17	2, 16	sylbi 121	. . . 4 |- ( 3o C_ suc ~P 1o -> 2o = ~P 1o )
18	17	eqcomd 2240	. . 3 |- ( 3o C_ suc ~P 1o -> ~P 1o = 2o )
19		exmidpweq 7171	. . 3 |- (EXMID <-> ~P 1o = 2o )
20	18, 19	sylibr 134	. 2 |- ( 3o C_ suc ~P 1o -> EXMID )
21	20	con3i 637	1 |- ( -. EXMID -> -. 3o C_ suc ~P 1o )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2205   C_ wss 3213   ~Pcpw 3671  EXMIDwem 4309   suc csuc 4488   omcom 4714   1oc1o 6642   2oc2o 6643   3oc3o 6644

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 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

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 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-uni 3917  df-int 3952  df-tr 4211  df-exmid 4310  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-1o 6649  df-2o 6650  df-3o 6651

This theorem is referenced by:  onntri45  7553