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Theorem 2oconcl 7628
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 4230 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2 difeq2 3755 . . . . . . . 8 (𝐴 = ∅ → (1𝑜𝐴) = (1𝑜 ∖ ∅))
3 dif0 3983 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2701 . . . . . . 7 (𝐴 = ∅ → (1𝑜𝐴) = 1𝑜)
5 difeq2 3755 . . . . . . . 8 (𝐴 = 1𝑜 → (1𝑜𝐴) = (1𝑜 ∖ 1𝑜))
6 difid 3981 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2701 . . . . . . 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 7612 . . . . . 6 1𝑜 ∈ On
12 difexg 4841 . . . . . 6 (1𝑜 ∈ On → (1𝑜𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1𝑜𝐴) ∈ V
1413elpr 4231 . . . 4 ((1𝑜𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
1510, 14sylibr 224 . . 3 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ {∅, 1𝑜})
16 df2o3 7618 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2741 . 2 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ 2𝑜)
1817, 16eleq2s 2748 1 (𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  c0 3948  {cpr 4212  Oncon0 5761  1𝑜c1o 7598  2𝑜c2o 7599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767  df-1o 7605  df-2o 7606
This theorem is referenced by:  efgmf  18172  efgmnvl  18173  efglem  18175  frgpuplem  18231
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