Theorem 3nsssucpw1 7166

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 6362	. . . . . 6 |- 3o = suc 2o
2	1	sseq1i 3154	. . . . 5 |- ( 3o C_ suc ~P 1o <-> suc 2o C_ suc ~P 1o )
3		1lt2o 6386	. . . . . . . . 9 |- 1o e. 2o
4		ssnel 4527	. . . . . . . . 9 |- ( 2o C_ 1o -> -. 1o e. 2o )
5	3, 4	mt2 630	. . . . . . . 8 |- -. 2o C_ 1o
6		2onn 6465	. . . . . . . . . 10 |- 2o e. om
7	6	elexi 2724	. . . . . . . . 9 |- 2o e. _V
8	7	elpw 3549	. . . . . . . 8 |- ( 2o e. ~P 1o <-> 2o C_ 1o )
9	5, 8	mtbir 661	. . . . . . 7 |- -. 2o e. ~P 1o
10	9	a1i 9	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> -. 2o e. ~P 1o )
11		sucssel 4384	. . . . . . . . 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 4363	. . . . . . . 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 719	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> ( 2o = ~P 1o \/ 2o e. ~P 1o ) )
16	10, 15	ecased 1331	. . . . 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 2163	. . 3 |- ( 3o C_ suc ~P 1o -> ~P 1o = 2o )
19		exmidpweq 6851	. . 3 |- (EXMID <-> ~P 1o = 2o )
20	18, 19	sylibr 133	. 2 |- ( 3o C_ suc ~P 1o -> EXMID )
21	20	con3i 622	1 |- ( -. EXMID -> -. 3o C_ suc ~P 1o )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 698   = wceq 1335   e. wcel 2128   C_ wss 3102   ~Pcpw 3543  EXMIDwem 4155   suc csuc 4325   omcom 4548   1oc1o 6353   2oc2o 6354   3oc3o 6355

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3773  df-int 3808  df-tr 4063  df-exmid 4156  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4549  df-1o 6360  df-2o 6361  df-3o 6362

This theorem is referenced by:  onntri45  7171