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Theorem 2oconcl 6053
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 3426 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2 difeq2 3084 . . . . . . . 8 (𝐴 = ∅ → (1𝑜𝐴) = (1𝑜 ∖ ∅))
3 dif0 3322 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2104 . . . . . . 7 (𝐴 = ∅ → (1𝑜𝐴) = 1𝑜)
5 difeq2 3084 . . . . . . . 8 (𝐴 = 1𝑜 → (1𝑜𝐴) = (1𝑜 ∖ 1𝑜))
6 difid 3320 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2104 . . . . . . 7 (𝐴 = 1𝑜 → (1𝑜𝐴) = ∅)
84, 7orim12i 686 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = 1𝑜 ∨ (1𝑜𝐴) = ∅))
98orcomd 658 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
101, 9syl 14 . . . 4 (𝐴 ∈ {∅, 1𝑜} → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
11 1on 6039 . . . . . 6 1𝑜 ∈ On
12 difexg 3926 . . . . . 6 (1𝑜 ∈ On → (1𝑜𝐴) ∈ V)
1311, 12ax-mp 7 . . . . 5 (1𝑜𝐴) ∈ V
1413elpr 3424 . . . 4 ((1𝑜𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
1510, 14sylibr 141 . . 3 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ {∅, 1𝑜})
16 df2o3 6045 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2147 . 2 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ 2𝑜)
1817, 16eleq2s 2148 1 (𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wcel 1409  Vcvv 2574  cdif 2942  c0 3252  {cpr 3404  Oncon0 4128  1𝑜c1o 6025  2𝑜c2o 6026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-tr 3883  df-iord 4131  df-on 4133  df-suc 4136  df-1o 6032  df-2o 6033
This theorem is referenced by: (None)
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