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Theorem 2oconcl 8106
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 4565 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2 difeq2 4072 . . . . . . . 8 (𝐴 = ∅ → (1o𝐴) = (1o ∖ ∅))
3 dif0 4308 . . . . . . . 8 (1o ∖ ∅) = 1o
42, 3syl6eq 2871 . . . . . . 7 (𝐴 = ∅ → (1o𝐴) = 1o)
5 difeq2 4072 . . . . . . . 8 (𝐴 = 1o → (1o𝐴) = (1o ∖ 1o))
6 difid 4306 . . . . . . . 8 (1o ∖ 1o) = ∅
75, 6syl6eq 2871 . . . . . . 7 (𝐴 = 1o → (1o𝐴) = ∅)
84, 7orim12i 905 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = 1o ∨ (1o𝐴) = ∅))
98orcomd 867 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
101, 9syl 17 . . . 4 (𝐴 ∈ {∅, 1o} → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
11 1on 8087 . . . . . 6 1o ∈ On
12 difexg 5207 . . . . . 6 (1o ∈ On → (1o𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1o𝐴) ∈ V
1413elpr 4566 . . . 4 ((1o𝐴) ∈ {∅, 1o} ↔ ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
1510, 14sylibr 236 . . 3 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ {∅, 1o})
16 df2o3 8095 . . 3 2o = {∅, 1o}
1715, 16eleqtrrdi 2922 . 2 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ 2o)
1817, 16eleq2s 2929 1 (𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 843   = wceq 1537  wcel 2114  Vcvv 3473  cdif 3910  c0 4269  {cpr 4545  Oncon0 6167  1oc1o 8073  2oc2o 8074
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 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-un 7439
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 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-ord 6170  df-on 6171  df-suc 6173  df-1o 8080  df-2o 8081
This theorem is referenced by:  efgmf  18818  efgmnvl  18819  efglem  18821  frgpuplem  18877
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