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Theorem 2oconcl 7867
 Description: Closure of the pair swapping function on 2o. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
2oconcl (𝐴 ∈ 2o → (1o𝐴) ∈ 2o)

Proof of Theorem 2oconcl
StepHypRef Expression
1 elpri 4420 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2 difeq2 3945 . . . . . . . 8 (𝐴 = ∅ → (1o𝐴) = (1o ∖ ∅))
3 dif0 4181 . . . . . . . 8 (1o ∖ ∅) = 1o
42, 3syl6eq 2830 . . . . . . 7 (𝐴 = ∅ → (1o𝐴) = 1o)
5 difeq2 3945 . . . . . . . 8 (𝐴 = 1o → (1o𝐴) = (1o ∖ 1o))
6 difid 4179 . . . . . . . 8 (1o ∖ 1o) = ∅
75, 6syl6eq 2830 . . . . . . 7 (𝐴 = 1o → (1o𝐴) = ∅)
84, 7orim12i 895 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = 1o ∨ (1o𝐴) = ∅))
98orcomd 860 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
101, 9syl 17 . . . 4 (𝐴 ∈ {∅, 1o} → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
11 1on 7850 . . . . . 6 1o ∈ On
12 difexg 5045 . . . . . 6 (1o ∈ On → (1o𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1o𝐴) ∈ V
1413elpr 4421 . . . 4 ((1o𝐴) ∈ {∅, 1o} ↔ ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
1510, 14sylibr 226 . . 3 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ {∅, 1o})
16 df2o3 7857 . . 3 2o = {∅, 1o}
1715, 16syl6eleqr 2870 . 2 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ 2o)
1817, 16eleq2s 2877 1 (𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 836   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ∖ cdif 3789  ∅c0 4141  {cpr 4400  Oncon0 5976  1oc1o 7836  2oc2o 7837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980  df-suc 5982  df-1o 7843  df-2o 7844 This theorem is referenced by:  efgmf  18510  efgmnvl  18511  efglem  18513  frgpuplem  18571
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