Theorem 2oconcl 8440

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 4606	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		difeq2 4074	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
3		dif0 4332	. . . . . . . 8 ⊢ (1o ∖ ∅) = 1o
4	2, 3	eqtrdi 2788	. . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
5		difeq2 4074	. . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o))
6		difid 4330	. . . . . . . 8 ⊢ (1o ∖ 1o) = ∅
7	5, 6	eqtrdi 2788	. . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅)
8	4, 7	orim12i 909	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅))
9	8	orcomd 872	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
10	1, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
11		1on 8419	. . . . . 6 ⊢ 1o ∈ On
12		difexg 5276	. . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1o ∖ 𝐴) ∈ V
14	13	elpr 4607	. . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
15	10, 14	sylibr 234	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o})
16		df2o3 8415	. . 3 ⊢ 2o = {∅, 1o}
17	15, 16	eleqtrrdi 2848	. 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o)
18	17, 16	eleq2s 2855	1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900  ∅c0 4287  {cpr 4584  Oncon0 6325  1_oc1o 8400  2_oc2o 8401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-suc 6331  df-1o 8407  df-2o 8408

This theorem is referenced by:  efgmf  19654  efgmnvl  19655  efglem  19657  frgpuplem  19713