Theorem 2oconcl 8111

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 4547	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2		difeq2 4044	. . . . . . . 8 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = (1o ∖ ∅))
3		dif0 4286	. . . . . . . 8 ⊢ (1o ∖ ∅) = 1o
4	2, 3	eqtrdi 2849	. . . . . . 7 ⊢ (𝐴 = ∅ → (1o ∖ 𝐴) = 1o)
5		difeq2 4044	. . . . . . . 8 ⊢ (𝐴 = 1o → (1o ∖ 𝐴) = (1o ∖ 1o))
6		difid 4284	. . . . . . . 8 ⊢ (1o ∖ 1o) = ∅
7	5, 6	eqtrdi 2849	. . . . . . 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 8092	. . . . . 6 ⊢ 1o ∈ On
12		difexg 5195	. . . . . 6 ⊢ (1o ∈ On → (1o ∖ 𝐴) ∈ V)
13	11, 12	ax-mp 5	. . . . 5 ⊢ (1o ∖ 𝐴) ∈ V
14	13	elpr 4548	. . . 4 ⊢ ((1o ∖ 𝐴) ∈ {∅, 1o} ↔ ((1o ∖ 𝐴) = ∅ ∨ (1o ∖ 𝐴) = 1o))
15	10, 14	sylibr 237	. . 3 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ {∅, 1o})
16		df2o3 8100	. . 3 ⊢ 2o = {∅, 1o}
17	15, 16	eleqtrrdi 2901	. 2 ⊢ (𝐴 ∈ {∅, 1o} → (1o ∖ 𝐴) ∈ 2o)
18	17, 16	eleq2s 2908	1 ⊢ (𝐴 ∈ 2o → (1o ∖ 𝐴) ∈ 2o)

Syntax hints:   → wi 4   ∨ wo 844   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  {cpr 4527  Oncon0 6159  1_oc1o 8078  2_oc2o 8079

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-suc 6165  df-1o 8085  df-2o 8086

This theorem is referenced by:  efgmf  18831  efgmnvl  18832  efglem  18834  frgpuplem  18890