Theorem 2oconcl 8333

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 4583	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		difeq2 4051	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
3		dif0 4306	. . . . . . . 8 ⊢ (1o ∖ ∅) = 1o
4	2, 3	eqtrdi 2794	. . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
5		difeq2 4051	. . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o))
6		difid 4304	. . . . . . . 8 ⊢ (1o ∖ 1o) = ∅
7	5, 6	eqtrdi 2794	. . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅)
8	4, 7	orim12i 906	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅))
9	8	orcomd 868	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
10	1, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
11		1on 8309	. . . . . 6 ⊢ 1o ∈ On
12		difexg 5251	. . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1o ∖ 𝐴) ∈ V
14	13	elpr 4584	. . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
15	10, 14	sylibr 233	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o})
16		df2o3 8305	. . 3 ⊢ 2o = {∅, 1o}
17	15, 16	eleqtrrdi 2850	. 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o)
18	17, 16	eleq2s 2857	1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884  ∅c0 4256  {cpr 4563  Oncon0 6266  1_oc1o 8290  2_oc2o 8291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-suc 6272  df-1o 8297  df-2o 8298

This theorem is referenced by:  efgmf  19319  efgmnvl  19320  efglem  19322  frgpuplem  19378