Theorem 2oconcl 6022

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 3398	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2		difeq2 3056	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ ∅))
3		dif0 3294	. . . . . . . 8 ⊢ (1𝑜 ∖ ∅) = 1𝑜
4	2, 3	syl6eq 2088	. . . . . . 7 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = 1𝑜)
5		difeq2 3056	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ 1𝑜))
6		difid 3292	. . . . . . . 8 ⊢ (1𝑜 ∖ 1𝑜) = ∅
7	5, 6	syl6eq 2088	. . . . . . 7 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = ∅)
8	4, 7	orim12i 676	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = 1𝑜 ∨ (1𝑜 ∖ 𝐴) = ∅))
9	8	orcomd 648	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
10	1, 9	syl 14	. . . 4 ⊢ (𝐴 ∈ {∅, 1𝑜} → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
11		1on 6008	. . . . . 6 ⊢ 1𝑜 ∈ On
12		difexg 3898	. . . . . 6 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 7	. . . . 5 ⊢ (1𝑜 ∖ 𝐴) ∈ V
14	13	elpr 3396	. . . 4 ⊢ ((1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
15	10, 14	sylibr 137	. . 3 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜})
16		df2o3 6014	. . 3 ⊢ 2𝑜 = {∅, 1𝑜}
17	15, 16	syl6eleqr 2131	. 2 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ 2𝑜)
18	17, 16	eleq2s 2132	1 ⊢ (𝐴 ∈ 2𝑜 → (1𝑜 ∖ 𝐴) ∈ 2𝑜)

Syntax hints:   → wi 4   ∨ wo 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∖ cdif 2914  ∅c0 3224  {cpr 3376  Oncon0 4100  1_𝑜c1o 5994  2_𝑜c2o 5995

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-1o 6001  df-2o 6002

This theorem is referenced by: (None)