Theorem 3nsssucpw1 7213

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 6397	. . . . . 6 |- 3o = suc 2o
2	1	sseq1i 3173	. . . . 5 |- ( 3o C_ suc ~P 1o <-> suc 2o C_ suc ~P 1o )
3		1lt2o 6421	. . . . . . . . 9 |- 1o e. 2o
4		ssnel 4553	. . . . . . . . 9 |- ( 2o C_ 1o -> -. 1o e. 2o )
5	3, 4	mt2 635	. . . . . . . 8 |- -. 2o C_ 1o
6		2onn 6500	. . . . . . . . . 10 |- 2o e. om
7	6	elexi 2742	. . . . . . . . 9 |- 2o e. _V
8	7	elpw 3572	. . . . . . . 8 |- ( 2o e. ~P 1o <-> 2o C_ 1o )
9	5, 8	mtbir 666	. . . . . . 7 |- -. 2o e. ~P 1o
10	9	a1i 9	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> -. 2o e. ~P 1o )
11		sucssel 4409	. . . . . . . . 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 4388	. . . . . . . 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 724	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> ( 2o = ~P 1o \/ 2o e. ~P 1o ) )
16	10, 15	ecased 1344	. . . . 5 |- ( suc 2o C_ suc ~P 1o -> 2o = ~P 1o )
17	2, 16	sylbi 120	. . . 4 |- ( 3o C_ suc ~P 1o -> 2o = ~P 1o )
18	17	eqcomd 2176	. . 3 |- ( 3o C_ suc ~P 1o -> ~P 1o = 2o )
19		exmidpweq 6887	. . 3 |- (EXMID <-> ~P 1o = 2o )
20	18, 19	sylibr 133	. 2 |- ( 3o C_ suc ~P 1o -> EXMID )
21	20	con3i 627	1 |- ( -. EXMID -> -. 3o C_ suc ~P 1o )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 703   = wceq 1348   e. wcel 2141   C_ wss 3121   ~Pcpw 3566  EXMIDwem 4180   suc csuc 4350   omcom 4574   1oc1o 6388   2oc2o 6389   3oc3o 6390

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-int 3832  df-tr 4088  df-exmid 4181  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-1o 6395  df-2o 6396  df-3o 6397

This theorem is referenced by:  onntri45  7218