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Theorem 2oconcl 7470
 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 4145 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2 difeq2 3684 . . . . . . . 8 (𝐴 = ∅ → (1𝑜𝐴) = (1𝑜 ∖ ∅))
3 dif0 3904 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2660 . . . . . . 7 (𝐴 = ∅ → (1𝑜𝐴) = 1𝑜)
5 difeq2 3684 . . . . . . . 8 (𝐴 = 1𝑜 → (1𝑜𝐴) = (1𝑜 ∖ 1𝑜))
6 difid 3902 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2660 . . . . . . 7 (𝐴 = 1𝑜 → (1𝑜𝐴) = ∅)
84, 7orim12i 537 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = 1𝑜 ∨ (1𝑜𝐴) = ∅))
98orcomd 402 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
101, 9syl 17 . . . 4 (𝐴 ∈ {∅, 1𝑜} → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
11 1on 7454 . . . . . 6 1𝑜 ∈ On
12 difexg 4735 . . . . . 6 (1𝑜 ∈ On → (1𝑜𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1𝑜𝐴) ∈ V
1413elpr 4146 . . . 4 ((1𝑜𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
1510, 14sylibr 223 . . 3 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ {∅, 1𝑜})
16 df2o3 7460 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2699 . 2 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ 2𝑜)
1817, 16eleq2s 2706 1 (𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  {cpr 4127  Oncon0 5640  1𝑜c1o 7440  2𝑜c2o 7441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-suc 5646  df-1o 7447  df-2o 7448 This theorem is referenced by:  efgmf  17949  efgmnvl  17950  efglem  17952  frgpuplem  18008
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