Theorem 2oconcl 6407

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 3599	. . . . 5 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		difeq2 3234	. . . . . . . 8 |- ( A = (/) -> ( 1o \ A ) = ( 1o \ (/) ) )
3		dif0 3479	. . . . . . . 8 |- ( 1o \ (/) ) = 1o
4	2, 3	eqtrdi 2215	. . . . . . 7 |- ( A = (/) -> ( 1o \ A ) = 1o )
5		difeq2 3234	. . . . . . . 8 |- ( A = 1o -> ( 1o \ A ) = ( 1o \ 1o ) )
6		difid 3477	. . . . . . . 8 |- ( 1o \ 1o ) = (/)
7	5, 6	eqtrdi 2215	. . . . . . 7 |- ( A = 1o -> ( 1o \ A ) = (/) )
8	4, 7	orim12i 749	. . . . . 6 |- ( ( A = (/) \/ A = 1o ) -> ( ( 1o \ A ) = 1o \/ ( 1o \ A ) = (/) ) )
9	8	orcomd 719	. . . . 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 6391	. . . . . 6 |- 1o e. On
12		difexg 4123	. . . . . 6 |- ( 1o e. On -> ( 1o \ A ) e. _V )
13	11, 12	ax-mp 5	. . . . 5 |- ( 1o \ A ) e. _V
14	13	elpr 3597	. . . 4 |- ( ( 1o \ A ) e. { (/) , 1o } <-> ( ( 1o \ A ) = (/) \/ ( 1o \ A ) = 1o ) )
15	10, 14	sylibr 133	. . 3 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. { (/) , 1o } )
16		df2o3 6398	. . 3 |- 2o = { (/) , 1o }
17	15, 16	eleqtrrdi 2260	. 2 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. 2o )
18	17, 16	eleq2s 2261	1 |- ( A e. 2o -> ( 1o \ A ) e. 2o )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1343   e. wcel 2136   _Vcvv 2726   \ cdif 3113   (/)c0 3409   {cpr 3577   Oncon0 4341   1oc1o 6377   2oc2o 6378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349  df-1o 6384  df-2o 6385

This theorem is referenced by:  ismkvnex  7119