Theorem 2oconcl 6045

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

Assertion

Ref	Expression
2oconcl	⊢ (𝐴 ∈ 2𝑜 → (1𝑜 ∖ 𝐴) ∈ 2𝑜)

Proof of Theorem 2oconcl

Step	Hyp	Ref	Expression
1		elpri 3421	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2		difeq2 3084	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ ∅))
3		dif0 3314	. . . . . . . 8 ⊢ (1𝑜 ∖ ∅) = 1𝑜
4	2, 3	syl6eq 2129	. . . . . . 7 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = 1𝑜)
5		difeq2 3084	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ 1𝑜))
6		difid 3312	. . . . . . . 8 ⊢ (1𝑜 ∖ 1𝑜) = ∅
7	5, 6	syl6eq 2129	. . . . . . 7 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = ∅)
8	4, 7	orim12i 708	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = 1𝑜 ∨ (1𝑜 ∖ 𝐴) = ∅))
9	8	orcomd 680	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
10	1, 9	syl 14	. . . 4 ⊢ (𝐴 ∈ {∅, 1𝑜} → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
11		1on 6031	. . . . . 6 ⊢ 1𝑜 ∈ On
12		difexg 3919	. . . . . 6 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 7	. . . . 5 ⊢ (1𝑜 ∖ 𝐴) ∈ V
14	13	elpr 3419	. . . 4 ⊢ ((1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
15	10, 14	sylibr 132	. . 3 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜})
16		df2o3 6037	. . 3 ⊢ 2𝑜 = {∅, 1𝑜}
17	15, 16	syl6eleqr 2172	. 2 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ 2𝑜)
18	17, 16	eleq2s 2173	1 ⊢ (𝐴 ∈ 2𝑜 → (1𝑜 ∖ 𝐴) ∈ 2𝑜)

Syntax hints:   → wi 4   ∨ wo 661   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∖ cdif 2970  ∅c0 3251  {cpr 3399  Oncon0 4118  1_𝑜c1o 6017  2_𝑜c2o 6018

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126  df-1o 6024  df-2o 6025

This theorem is referenced by: (None)