Theorem 3nelsucpw1 7190

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 6410	. . . . 5 |- 1o e. 2o
2		elelsuc 4387	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6386	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2242	. . 3 |- 1o e. 3o
6		ssnel 4546	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 630	. 2 |- -. 3o C_ 1o
8		pw1ne3 7186	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2382	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4381	. . . 4 |- ( 3o e. suc ~P 1o -> ( 3o e. ~P 1o \/ 3o = ~P 1o ) )
12	10, 11	ecased 1339	. . 3 |- ( 3o e. suc ~P 1o -> 3o e. ~P 1o )
13	12	elpwid 3570	. 2 |- ( 3o e. suc ~P 1o -> 3o C_ 1o )
14	7, 13	mto 652	1 |- -. 3o e. suc ~P 1o

Syntax hints:   -. wn 3   = wceq 1343   e. wcel 2136   C_ wss 3116   ~Pcpw 3559   suc csuc 4343   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-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-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:  onntri35  7193