Theorem 3nelsucpw1 7557

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 6688	. . . . 5 |- 1o e. 2o
2		elelsuc 4535	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6662	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2310	. . 3 |- 1o e. 3o
6		ssnel 4696	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 645	. 2 |- -. 3o C_ 1o
8		pw1ne3 7553	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2460	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4529	. . . 4 |- ( 3o e. suc ~P 1o -> ( 3o e. ~P 1o \/ 3o = ~P 1o ) )
12	10, 11	ecased 1386	. . 3 |- ( 3o e. suc ~P 1o -> 3o e. ~P 1o )
13	12	elpwid 3685	. 2 |- ( 3o e. suc ~P 1o -> 3o C_ 1o )
14	7, 13	mto 668	1 |- -. 3o e. suc ~P 1o

Syntax hints:   -. wn 3   = wceq 1398   e. wcel 2205   C_ wss 3214   ~Pcpw 3674   suc csuc 4491   1oc1o 6653   2oc2o 6654   3oc3o 6655

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-tr 4214  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-1o 6660  df-2o 6661  df-3o 6662

This theorem is referenced by:  onntri35  7560