Theorem 3nelsucpw1 7583

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 6705	. . . . 5 |- 1o e. 2o
2		elelsuc 4549	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6679	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2314	. . 3 |- 1o e. 3o
6		ssnel 4711	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 649	. 2 |- -. 3o C_ 1o
8		pw1ne3 7579	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2466	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4543	. . . 4 |- ( 3o e. suc ~P 1o -> ( 3o e. ~P 1o \/ 3o = ~P 1o ) )
12	10, 11	ecased 1390	. . 3 |- ( 3o e. suc ~P 1o -> 3o e. ~P 1o )
13	12	elpwid 3696	. 2 |- ( 3o e. suc ~P 1o -> 3o C_ 1o )
14	7, 13	mto 672	1 |- -. 3o e. suc ~P 1o

Syntax hints:   -. wn 3   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3685   suc csuc 4505   1oc1o 6670   2oc2o 6671   3oc3o 6672

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-int 3966  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-1o 6677  df-2o 6678  df-3o 6679

This theorem is referenced by:  onntri35  7586