Theorem 2oconcl 6398

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 3593	. . . . 5 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		difeq2 3229	. . . . . . . 8 |- ( A = (/) -> ( 1o \ A ) = ( 1o \ (/) ) )
3		dif0 3474	. . . . . . . 8 |- ( 1o \ (/) ) = 1o
4	2, 3	eqtrdi 2213	. . . . . . 7 |- ( A = (/) -> ( 1o \ A ) = 1o )
5		difeq2 3229	. . . . . . . 8 |- ( A = 1o -> ( 1o \ A ) = ( 1o \ 1o ) )
6		difid 3472	. . . . . . . 8 |- ( 1o \ 1o ) = (/)
7	5, 6	eqtrdi 2213	. . . . . . 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 6382	. . . . . 6 |- 1o e. On
12		difexg 4117	. . . . . 6 |- ( 1o e. On -> ( 1o \ A ) e. _V )
13	11, 12	ax-mp 5	. . . . 5 |- ( 1o \ A ) e. _V
14	13	elpr 3591	. . . 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 6389	. . 3 |- 2o = { (/) , 1o }
17	15, 16	eleqtrrdi 2258	. 2 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. 2o )
18	17, 16	eleq2s 2259	1 |- ( A e. 2o -> ( 1o \ A ) e. 2o )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1342   e. wcel 2135   _Vcvv 2721   \ cdif 3108   (/)c0 3404   {cpr 3571   Oncon0 4335   1oc1o 6368   2oc2o 6369

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-uni 3784  df-tr 4075  df-iord 4338  df-on 4340  df-suc 4343  df-1o 6375  df-2o 6376

This theorem is referenced by:  ismkvnex  7110