Theorem 2oconcl 6525

Description: Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)

Assertion

Ref	Expression
2oconcl	|- ( A e. 2o -> ( 1o \ A ) e. 2o )

Proof of Theorem 2oconcl

Step	Hyp	Ref	Expression
1		elpri 3656	. . . . 5 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		difeq2 3285	. . . . . . . 8 |- ( A = (/) -> ( 1o \ A ) = ( 1o \ (/) ) )
3		dif0 3531	. . . . . . . 8 |- ( 1o \ (/) ) = 1o
4	2, 3	eqtrdi 2254	. . . . . . 7 |- ( A = (/) -> ( 1o \ A ) = 1o )
5		difeq2 3285	. . . . . . . 8 |- ( A = 1o -> ( 1o \ A ) = ( 1o \ 1o ) )
6		difid 3529	. . . . . . . 8 |- ( 1o \ 1o ) = (/)
7	5, 6	eqtrdi 2254	. . . . . . 7 |- ( A = 1o -> ( 1o \ A ) = (/) )
8	4, 7	orim12i 761	. . . . . 6 |- ( ( A = (/) \/ A = 1o ) -> ( ( 1o \ A ) = 1o \/ ( 1o \ A ) = (/) ) )
9	8	orcomd 731	. . . . 5 |- ( ( A = (/) \/ A = 1o ) -> ( ( 1o \ A ) = (/) \/ ( 1o \ A ) = 1o ) )
10	1, 9	syl 14	. . . 4 |- ( A e. { (/) , 1o } -> ( ( 1o \ A ) = (/) \/ ( 1o \ A ) = 1o ) )
11		1on 6509	. . . . . 6 |- 1o e. On
12		difexg 4185	. . . . . 6 |- ( 1o e. On -> ( 1o \ A ) e. _V )
13	11, 12	ax-mp 5	. . . . 5 |- ( 1o \ A ) e. _V
14	13	elpr 3654	. . . 4 |- ( ( 1o \ A ) e. { (/) , 1o } <-> ( ( 1o \ A ) = (/) \/ ( 1o \ A ) = 1o ) )
15	10, 14	sylibr 134	. . 3 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. { (/) , 1o } )
16		df2o3 6516	. . 3 |- 2o = { (/) , 1o }
17	15, 16	eleqtrrdi 2299	. 2 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. 2o )
18	17, 16	eleq2s 2300	1 |- ( A e. 2o -> ( 1o \ A ) e. 2o )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2176   _Vcvv 2772   \ cdif 3163   (/)c0 3460   {cpr 3634   Oncon0 4410   1oc1o 6495   2oc2o 6496

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 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  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-tr 4143  df-iord 4413  df-on 4415  df-suc 4418  df-1o 6502  df-2o 6503

This theorem is referenced by:  ismkvnex  7257