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Theorem 2oconcl 6022
 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 3398 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
2 difeq2 3056 . . . . . . . 8 (𝐴 = ∅ → (1𝑜𝐴) = (1𝑜 ∖ ∅))
3 dif0 3294 . . . . . . . 8 (1𝑜 ∖ ∅) = 1𝑜
42, 3syl6eq 2088 . . . . . . 7 (𝐴 = ∅ → (1𝑜𝐴) = 1𝑜)
5 difeq2 3056 . . . . . . . 8 (𝐴 = 1𝑜 → (1𝑜𝐴) = (1𝑜 ∖ 1𝑜))
6 difid 3292 . . . . . . . 8 (1𝑜 ∖ 1𝑜) = ∅
75, 6syl6eq 2088 . . . . . . 7 (𝐴 = 1𝑜 → (1𝑜𝐴) = ∅)
84, 7orim12i 676 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = 1𝑜 ∨ (1𝑜𝐴) = ∅))
98orcomd 648 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
101, 9syl 14 . . . 4 (𝐴 ∈ {∅, 1𝑜} → ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
11 1on 6008 . . . . . 6 1𝑜 ∈ On
12 difexg 3898 . . . . . 6 (1𝑜 ∈ On → (1𝑜𝐴) ∈ V)
1311, 12ax-mp 7 . . . . 5 (1𝑜𝐴) ∈ V
1413elpr 3396 . . . 4 ((1𝑜𝐴) ∈ {∅, 1𝑜} ↔ ((1𝑜𝐴) = ∅ ∨ (1𝑜𝐴) = 1𝑜))
1510, 14sylibr 137 . . 3 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ {∅, 1𝑜})
16 df2o3 6014 . . 3 2𝑜 = {∅, 1𝑜}
1715, 16syl6eleqr 2131 . 2 (𝐴 ∈ {∅, 1𝑜} → (1𝑜𝐴) ∈ 2𝑜)
1817, 16eleq2s 2132 1 (𝐴 ∈ 2𝑜 → (1𝑜𝐴) ∈ 2𝑜)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∖ cdif 2914  ∅c0 3224  {cpr 3376  Oncon0 4100  1𝑜c1o 5994  2𝑜c2o 5995 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108  df-1o 6001  df-2o 6002 This theorem is referenced by: (None)
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