Theorem 3nelsucpw1 7251

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 6461	. . . . 5 |- 1o e. 2o
2		elelsuc 4424	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6437	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2265	. . 3 |- 1o e. 3o
6		ssnel 4583	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 641	. 2 |- -. 3o C_ 1o
8		pw1ne3 7247	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2406	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4418	. . . 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 3601	. 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 2160   C_ wss 3144   ~Pcpw 3590   suc csuc 4380   1oc1o 6428   2oc2o 6429   3oc3o 6430

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-tr 4117  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-1o 6435  df-2o 6436  df-3o 6437

This theorem is referenced by:  onntri35  7254