Theorem 2oconcl 8413

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 4598	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		difeq2 4068	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
3		dif0 4326	. . . . . . . 8 ⊢ (1o ∖ ∅) = 1o
4	2, 3	eqtrdi 2781	. . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
5		difeq2 4068	. . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o))
6		difid 4324	. . . . . . . 8 ⊢ (1o ∖ 1o) = ∅
7	5, 6	eqtrdi 2781	. . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅)
8	4, 7	orim12i 908	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅))
9	8	orcomd 871	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
10	1, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
11		1on 8392	. . . . . 6 ⊢ 1o ∈ On
12		difexg 5265	. . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1o ∖ 𝐴) ∈ V
14	13	elpr 4599	. . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
15	10, 14	sylibr 234	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o})
16		df2o3 8388	. . 3 ⊢ 2o = {∅, 1o}
17	15, 16	eleqtrrdi 2840	. 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o)
18	17, 16	eleq2s 2847	1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1541   ∈ wcel 2110  Vcvv 3434   ∖ cdif 3897  ∅c0 4281  {cpr 4576  Oncon0 6302  1_oc1o 8373  2_oc2o 8374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6305  df-on 6306  df-suc 6308  df-1o 8380  df-2o 8381

This theorem is referenced by:  efgmf  19618  efgmnvl  19619  efglem  19621  frgpuplem  19677