Theorem 3nelsucpw1 7233

Description: Three is not an element of the successor of the power set of 1o. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
3nelsucpw1	|- -. 3o e. suc ~P 1o

Proof of Theorem 3nelsucpw1

Step	Hyp	Ref	Expression
1		1lt2o 6443	. . . . 5 |- 1o e. 2o
2		elelsuc 4410	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6419	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2253	. . 3 |- 1o e. 3o
6		ssnel 4569	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 640	. 2 |- -. 3o C_ 1o
8		pw1ne3 7229	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2393	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4404	. . . 4 |- ( 3o e. suc ~P 1o -> ( 3o e. ~P 1o \/ 3o = ~P 1o ) )
12	10, 11	ecased 1349	. . 3 |- ( 3o e. suc ~P 1o -> 3o e. ~P 1o )
13	12	elpwid 3587	. 2 |- ( 3o e. suc ~P 1o -> 3o C_ 1o )
14	7, 13	mto 662	1 |- -. 3o e. suc ~P 1o

Syntax hints:   -. wn 3   = wceq 1353   e. wcel 2148   C_ wss 3130   ~Pcpw 3576   suc csuc 4366   1oc1o 6410   2oc2o 6411   3oc3o 6412

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-int 3846  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-1o 6417  df-2o 6418  df-3o 6419

This theorem is referenced by:  onntri35  7236