Theorem pw1ne3 7438

Description: The power set of 1o is not three. (Contributed by James E. Hanson and Jim Kingdon, 30-Jul-2024.)

Assertion

Ref	Expression
pw1ne3	|- ~P 1o =/= 3o

Proof of Theorem pw1ne3

Step	Hyp	Ref	Expression
1		1lt2o 6605	. . . . 5 |- 1o e. 2o
2		ssnel 4665	. . . . 5 |- ( 2o C_ 1o -> -. 1o e. 2o )
3	1, 2	mt2 643	. . . 4 |- -. 2o C_ 1o
4		2onn 6684	. . . . . 6 |- 2o e. om
5	4	elexi 2813	. . . . 5 |- 2o e. _V
6	5	elpw 3656	. . . 4 |- ( 2o e. ~P 1o <-> 2o C_ 1o )
7	3, 6	mtbir 675	. . 3 |- -. 2o e. ~P 1o
8	5	sucid 4512	. . . . 5 |- 2o e. suc 2o
9		df-3o 6579	. . . . 5 |- 3o = suc 2o
10	8, 9	eleqtrri 2305	. . . 4 |- 2o e. 3o
11		eleq2 2293	. . . 4 |- ( ~P 1o = 3o -> ( 2o e. ~P 1o <-> 2o e. 3o ) )
12	10, 11	mpbiri 168	. . 3 |- ( ~P 1o = 3o -> 2o e. ~P 1o )
13	7, 12	mto 666	. 2 |- -. ~P 1o = 3o
14	13	neir 2403	1 |- ~P 1o =/= 3o

Syntax hints:   = wceq 1395   e. wcel 2200   =/= wne 2400   C_ wss 3198   ~Pcpw 3650   suc csuc 4460   omcom 4686   1oc1o 6570   2oc2o 6571   3oc3o 6572

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-int 3927  df-tr 4186  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-1o 6577  df-2o 6578  df-3o 6579

This theorem is referenced by:  3nelsucpw1  7442