Theorem 2oconcl 6203

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 3469	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2		difeq2 3112	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ ∅))
3		dif0 3353	. . . . . . . 8 ⊢ (1𝑜 ∖ ∅) = 1𝑜
4	2, 3	syl6eq 2136	. . . . . . 7 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = 1𝑜)
5		difeq2 3112	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ 1𝑜))
6		difid 3351	. . . . . . . 8 ⊢ (1𝑜 ∖ 1𝑜) = ∅
7	5, 6	syl6eq 2136	. . . . . . 7 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = ∅)
8	4, 7	orim12i 711	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = 1𝑜 ∨ (1𝑜 ∖ 𝐴) = ∅))
9	8	orcomd 683	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
10	1, 9	syl 14	. . . 4 ⊢ (𝐴 ∈ {∅, 1𝑜} → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
11		1on 6188	. . . . . 6 ⊢ 1𝑜 ∈ On
12		difexg 3980	. . . . . 6 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 7	. . . . 5 ⊢ (1𝑜 ∖ 𝐴) ∈ V
14	13	elpr 3467	. . . 4 ⊢ ((1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
15	10, 14	sylibr 132	. . 3 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜})
16		df2o3 6195	. . 3 ⊢ 2𝑜 = {∅, 1𝑜}
17	15, 16	syl6eleqr 2181	. 2 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ 2𝑜)
18	17, 16	eleq2s 2182	1 ⊢ (𝐴 ∈ 2𝑜 → (1𝑜 ∖ 𝐴) ∈ 2𝑜)

Syntax hints:   → wi 4   ∨ wo 664   = wceq 1289   ∈ wcel 1438  Vcvv 2619   ∖ cdif 2996  ∅c0 3286  {cpr 3447  Oncon0 4190  1_𝑜c1o 6174  2_𝑜c2o 6175

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-uni 3654  df-tr 3937  df-iord 4193  df-on 4195  df-suc 4198  df-1o 6181  df-2o 6182

This theorem is referenced by: (None)