Theorem 3nsssucpw1 7192

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 6386	. . . . . 6 |- 3o = suc 2o
2	1	sseq1i 3168	. . . . 5 |- ( 3o C_ suc ~P 1o <-> suc 2o C_ suc ~P 1o )
3		1lt2o 6410	. . . . . . . . 9 |- 1o e. 2o
4		ssnel 4546	. . . . . . . . 9 |- ( 2o C_ 1o -> -. 1o e. 2o )
5	3, 4	mt2 630	. . . . . . . 8 |- -. 2o C_ 1o
6		2onn 6489	. . . . . . . . . 10 |- 2o e. om
7	6	elexi 2738	. . . . . . . . 9 |- 2o e. _V
8	7	elpw 3565	. . . . . . . 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 4402	. . . . . . . . 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 4381	. . . . . . . 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 1339	. . . . 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 2171	. . 3 |- ( 3o C_ suc ~P 1o -> ~P 1o = 2o )
19		exmidpweq 6875	. . 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 1343   e. wcel 2136   C_ wss 3116   ~Pcpw 3559  EXMIDwem 4173   suc csuc 4343   omcom 4567   1oc1o 6377   2oc2o 6378   3oc3o 6379

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-int 3825  df-tr 4081  df-exmid 4174  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-1o 6384  df-2o 6385  df-3o 6386

This theorem is referenced by:  onntri45  7197