Theorem 3nsssucpw1 7303

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 6476	. . . . . 6 |- 3o = suc 2o
2	1	sseq1i 3209	. . . . 5 |- ( 3o C_ suc ~P 1o <-> suc 2o C_ suc ~P 1o )
3		1lt2o 6500	. . . . . . . . 9 |- 1o e. 2o
4		ssnel 4605	. . . . . . . . 9 |- ( 2o C_ 1o -> -. 1o e. 2o )
5	3, 4	mt2 641	. . . . . . . 8 |- -. 2o C_ 1o
6		2onn 6579	. . . . . . . . . 10 |- 2o e. om
7	6	elexi 2775	. . . . . . . . 9 |- 2o e. _V
8	7	elpw 3611	. . . . . . . 8 |- ( 2o e. ~P 1o <-> 2o C_ 1o )
9	5, 8	mtbir 672	. . . . . . 7 |- -. 2o e. ~P 1o
10	9	a1i 9	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> -. 2o e. ~P 1o )
11		sucssel 4459	. . . . . . . . 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 4438	. . . . . . . 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 730	. . . . . 6 |- ( suc 2o C_ suc ~P 1o -> ( 2o = ~P 1o \/ 2o e. ~P 1o ) )
16	10, 15	ecased 1360	. . . . 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 2202	. . 3 |- ( 3o C_ suc ~P 1o -> ~P 1o = 2o )
19		exmidpweq 6970	. . 3 |- (EXMID <-> ~P 1o = 2o )
20	18, 19	sylibr 134	. 2 |- ( 3o C_ suc ~P 1o -> EXMID )
21	20	con3i 633	1 |- ( -. EXMID -> -. 3o C_ suc ~P 1o )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 709   = wceq 1364   e. wcel 2167   C_ wss 3157   ~Pcpw 3605  EXMIDwem 4227   suc csuc 4400   omcom 4626   1oc1o 6467   2oc2o 6468   3oc3o 6469

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-tr 4132  df-exmid 4228  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-1o 6474  df-2o 6475  df-3o 6476

This theorem is referenced by:  onntri45  7308