Theorem 2oconcl 8130

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 4591	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		difeq2 4095	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
3		dif0 4334	. . . . . . . 8 ⊢ (1o ∖ ∅) = 1o
4	2, 3	syl6eq 2874	. . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
5		difeq2 4095	. . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o))
6		difid 4332	. . . . . . . 8 ⊢ (1o ∖ 1o) = ∅
7	5, 6	syl6eq 2874	. . . . . . 7 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = ∅)
8	4, 7	orim12i 905	. . . . . 6 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = 1o ∨ (1o ∖ 𝐴) = ∅))
9	8	orcomd 867	. . . . 5 ⊢ ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
10	1, 9	syl 17	. . . 4 ⊢ (𝐴 ∈ {∅, 1o} → ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
11		1on 8111	. . . . . 6 ⊢ 1o ∈ On
12		difexg 5233	. . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1o ∖ 𝐴) ∈ V
14	13	elpr 4592	. . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
15	10, 14	sylibr 236	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o})
16		df2o3 8119	. . 3 ⊢ 2o = {∅, 1o}
17	15, 16	eleqtrrdi 2926	. 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o)
18	17, 16	eleq2s 2933	1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)

Syntax hints:   → wi 4   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Vcvv 3496   ∖ cdif 3935  ∅c0 4293  {cpr 4571  Oncon0 6193  1_oc1o 8097  2_oc2o 8098

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-suc 6199  df-1o 8104  df-2o 8105

This theorem is referenced by:  efgmf  18841  efgmnvl  18842  efglem  18844  frgpuplem  18900