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Theorem 2oconcl 8501
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 4645 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2 difeq2 4111 . . . . . . . 8 (𝐴 = ∅ → (1o𝐴) = (1o ∖ ∅))
3 dif0 4367 . . . . . . . 8 (1o ∖ ∅) = 1o
42, 3eqtrdi 2782 . . . . . . 7 (𝐴 = ∅ → (1o𝐴) = 1o)
5 difeq2 4111 . . . . . . . 8 (𝐴 = 1o → (1o𝐴) = (1o ∖ 1o))
6 difid 4365 . . . . . . . 8 (1o ∖ 1o) = ∅
75, 6eqtrdi 2782 . . . . . . 7 (𝐴 = 1o → (1o𝐴) = ∅)
84, 7orim12i 905 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = 1o ∨ (1o𝐴) = ∅))
98orcomd 868 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
101, 9syl 17 . . . 4 (𝐴 ∈ {∅, 1o} → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
11 1on 8476 . . . . . 6 1o ∈ On
12 difexg 5320 . . . . . 6 (1o ∈ On → (1o𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1o𝐴) ∈ V
1413elpr 4646 . . . 4 ((1o𝐴) ∈ {∅, 1o} ↔ ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
1510, 14sylibr 233 . . 3 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ {∅, 1o})
16 df2o3 8472 . . 3 2o = {∅, 1o}
1715, 16eleqtrrdi 2838 . 2 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ 2o)
1817, 16eleq2s 2845 1 (𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1533  wcel 2098  Vcvv 3468  cdif 3940  c0 4317  {cpr 4625  Oncon0 6357  1oc1o 8457  2oc2o 8458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-suc 6363  df-1o 8464  df-2o 8465
This theorem is referenced by:  efgmf  19631  efgmnvl  19632  efglem  19634  frgpuplem  19690
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