Theorem 3nelsucpw1 7348

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 6530	. . . . 5 |- 1o e. 2o
2		elelsuc 4457	. . . . 5 |- ( 1o e. 2o -> 1o e. suc 2o )
3	1, 2	ax-mp 5	. . . 4 |- 1o e. suc 2o
4		df-3o 6506	. . . 4 |- 3o = suc 2o
5	3, 4	eleqtrri 2281	. . 3 |- 1o e. 3o
6		ssnel 4618	. . 3 |- ( 3o C_ 1o -> -. 1o e. 3o )
7	5, 6	mt2 641	. 2 |- -. 3o C_ 1o
8		pw1ne3 7344	. . . . . 6 |- ~P 1o =/= 3o
9	8	nesymi 2422	. . . . 5 |- -. 3o = ~P 1o
10	9	a1i 9	. . . 4 |- ( 3o e. suc ~P 1o -> -. 3o = ~P 1o )
11		elsuci 4451	. . . 4 |- ( 3o e. suc ~P 1o -> ( 3o e. ~P 1o \/ 3o = ~P 1o ) )
12	10, 11	ecased 1362	. . 3 |- ( 3o e. suc ~P 1o -> 3o e. ~P 1o )
13	12	elpwid 3627	. 2 |- ( 3o e. suc ~P 1o -> 3o C_ 1o )
14	7, 13	mto 664	1 |- -. 3o e. suc ~P 1o

Syntax hints:   -. wn 3   = wceq 1373   e. wcel 2176   C_ wss 3166   ~Pcpw 3616   suc csuc 4413   1oc1o 6497   2oc2o 6498   3oc3o 6499

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4144  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640  df-1o 6504  df-2o 6505  df-3o 6506

This theorem is referenced by:  onntri35  7351