Theorem 2oconcl 6543

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 3661	. . . . 5 |- ( A e. { (/) , 1o } -> ( A = (/) \/ A = 1o ) )
2		difeq2 3289	. . . . . . . 8 |- ( A = (/) -> ( 1o \ A ) = ( 1o \ (/) ) )
3		dif0 3535	. . . . . . . 8 |- ( 1o \ (/) ) = 1o
4	2, 3	eqtrdi 2255	. . . . . . 7 |- ( A = (/) -> ( 1o \ A ) = 1o )
5		difeq2 3289	. . . . . . . 8 |- ( A = 1o -> ( 1o \ A ) = ( 1o \ 1o ) )
6		difid 3533	. . . . . . . 8 |- ( 1o \ 1o ) = (/)
7	5, 6	eqtrdi 2255	. . . . . . 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 6527	. . . . . 6 |- 1o e. On
12		difexg 4196	. . . . . 6 |- ( 1o e. On -> ( 1o \ A ) e. _V )
13	11, 12	ax-mp 5	. . . . 5 |- ( 1o \ A ) e. _V
14	13	elpr 3659	. . . 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 6534	. . 3 |- 2o = { (/) , 1o }
17	15, 16	eleqtrrdi 2300	. 2 |- ( A e. { (/) , 1o } -> ( 1o \ A ) e. 2o )
18	17, 16	eleq2s 2301	1 |- ( A e. 2o -> ( 1o \ A ) e. 2o )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2177   _Vcvv 2773   \ cdif 3167   (/)c0 3464   {cpr 3639   Oncon0 4423   1oc1o 6513   2oc2o 6514

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-uni 3860  df-tr 4154  df-iord 4426  df-on 4428  df-suc 4431  df-1o 6520  df-2o 6521

This theorem is referenced by:  ismkvnex  7278