Theorem pw1ne3 7491

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 6653	. . . . 5 |- 1o e. 2o
2		ssnel 4673	. . . . 5 |- ( 2o C_ 1o -> -. 1o e. 2o )
3	1, 2	mt2 645	. . . 4 |- -. 2o C_ 1o
4		2onn 6732	. . . . . 6 |- 2o e. om
5	4	elexi 2816	. . . . 5 |- 2o e. _V
6	5	elpw 3662	. . . 4 |- ( 2o e. ~P 1o <-> 2o C_ 1o )
7	3, 6	mtbir 678	. . 3 |- -. 2o e. ~P 1o
8	5	sucid 4520	. . . . 5 |- 2o e. suc 2o
9		df-3o 6627	. . . . 5 |- 3o = suc 2o
10	8, 9	eleqtrri 2307	. . . 4 |- 2o e. 3o
11		eleq2 2295	. . . 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 668	. 2 |- -. ~P 1o = 3o
14	13	neir 2406	1 |- ~P 1o =/= 3o

Syntax hints:   = wceq 1398   e. wcel 2202   =/= wne 2403   C_ wss 3201   ~Pcpw 3656   suc csuc 4468   omcom 4694   1oc1o 6618   2oc2o 6619   3oc3o 6620

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-1o 6625  df-2o 6626  df-3o 6627

This theorem is referenced by:  3nelsucpw1  7495