Theorem 2oconcl 7583

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 4197	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2		difeq2 3722	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ ∅))
3		dif0 3950	. . . . . . . 8 ⊢ (1𝑜 ∖ ∅) = 1𝑜
4	2, 3	syl6eq 2672	. . . . . . 7 ⊢ (𝐴 = ∅ → (1𝑜 ∖ 𝐴) = 1𝑜)
5		difeq2 3722	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = (1𝑜 ∖ 1𝑜))
6		difid 3948	. . . . . . . 8 ⊢ (1𝑜 ∖ 1𝑜) = ∅
7	5, 6	syl6eq 2672	. . . . . . 7 ⊢ (𝐴 = 1𝑜 → (1𝑜 ∖ 𝐴) = ∅)
8	4, 7	orim12i 538	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = 1𝑜 ∨ (1𝑜 ∖ 𝐴) = ∅))
9	8	orcomd 403	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
10	1, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ {∅, 1𝑜} → ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
11		1on 7567	. . . . . 6 ⊢ 1𝑜 ∈ On
12		difexg 4808	. . . . . 6 ⊢ (1𝑜 ∈ On → (1𝑜 ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1𝑜 ∖ 𝐴) ∈ V
14	13	elpr 4198	. . . 4 ⊢ ((1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜 ∖ 𝐴) = ∅ ∨ (1𝑜 ∖ 𝐴) = 1𝑜))
15	10, 14	sylibr 224	. . 3 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ {∅, 1𝑜})
16		df2o3 7573	. . 3 ⊢ 2𝑜 = {∅, 1𝑜}
17	15, 16	syl6eleqr 2712	. 2 ⊢ (𝐴 ∈ {∅, 1𝑜} → (1𝑜 ∖ 𝐴) ∈ 2𝑜)
18	17, 16	eleq2s 2719	1 ⊢ (𝐴 ∈ 2𝑜 → (1𝑜 ∖ 𝐴) ∈ 2𝑜)

Syntax hints:   → wi 4   ∨ wo 383   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  {cpr 4179  Oncon0 5723  1_𝑜c1o 7553  2_𝑜c2o 7554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729  df-1o 7560  df-2o 7561

This theorem is referenced by:  efgmf  18126  efgmnvl  18127  efglem  18129  frgpuplem  18185