Theorem 3nelsucpw1 7296

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 6497	. . . . 5 |- 1o e. 2o
2		elelsuc 4441	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6473	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2269	. . 3 |- 1o e. 3o
6		ssnel 4602	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 641	. 2 |- -. 3o C_ 1o
8		pw1ne3 7292	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2410	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4435	. . . 4 |- ( 3o e. suc ~P 1o -> ( 3o e. ~P 1o \/ 3o = ~P 1o ) )
12	10, 11	ecased 1360	. . 3 |- ( 3o e. suc ~P 1o -> 3o e. ~P 1o )
13	12	elpwid 3613	. 2 |- ( 3o e. suc ~P 1o -> 3o C_ 1o )
14	7, 13	mto 663	1 |- -. 3o e. suc ~P 1o

Syntax hints:   -. wn 3   = wceq 1364   e. wcel 2164   C_ wss 3154   ~Pcpw 3602   suc csuc 4397   1oc1o 6464   2oc2o 6465   3oc3o 6466

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-1o 6471  df-2o 6472  df-3o 6473

This theorem is referenced by:  onntri35  7299