Theorem 2oconcl 6524

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 3655	. . . . 5 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		difeq2 3284	. . . . . . . 8 |- ( A = (/) -> ( 1o \ A ) = ( 1o \ (/) ) )
3		dif0 3530	. . . . . . . 8 |- ( 1o \ (/) ) = 1o
4	2, 3	eqtrdi 2253	. . . . . . 7 |- ( A = (/) -> ( 1o \ A ) = 1o )
5		difeq2 3284	. . . . . . . 8 |- ( A = 1o -> ( 1o \ A ) = ( 1o \ 1o ) )
6		difid 3528	. . . . . . . 8 |- ( 1o \ 1o ) = (/)
7	5, 6	eqtrdi 2253	. . . . . . 7 |- ( A = 1o -> ( 1o \ A ) = (/) )
8	4, 7	orim12i 760	. . . . . 6 |- ( ( A = (/) \/ A = 1o ) -> ( ( 1o \ A ) = 1o \/ ( 1o \ A ) = (/) ) )
9	8	orcomd 730	. . . . 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 6508	. . . . . 6 |- 1o e. On
12		difexg 4184	. . . . . 6 |- ( 1o e. On -> ( 1o \ A ) e. _V )
13	11, 12	ax-mp 5	. . . . 5 |- ( 1o \ A ) e. _V
14	13	elpr 3653	. . . 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 6515	. . 3 |- 2o = { (/) , 1o }
17	15, 16	eleqtrrdi 2298	. 2 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. 2o )
18	17, 16	eleq2s 2299	1 |- ( A e. 2o -> ( 1o \ A ) e. 2o )

Syntax hints:   -> wi 4   \/ wo 709   = wceq 1372   e. wcel 2175   _Vcvv 2771   \ cdif 3162   (/)c0 3459   {cpr 3633   Oncon0 4409   1oc1o 6494   2oc2o 6495

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-uni 3850  df-tr 4142  df-iord 4412  df-on 4414  df-suc 4417  df-1o 6501  df-2o 6502

This theorem is referenced by:  ismkvnex  7256