Theorem 3nelsucpw1 7301

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 6500	. . . . 5 |- 1o e. 2o
2		elelsuc 4444	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6476	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2272	. . 3 |- 1o e. 3o
6		ssnel 4605	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 641	. 2 |- -. 3o C_ 1o
8		pw1ne3 7297	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2413	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4438	. . . 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 3616	. 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 2167   C_ wss 3157   ~Pcpw 3605   suc csuc 4400   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-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-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:  onntri35  7304