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Theorem 2oconcl 6465
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 3630 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2 difeq2 3262 . . . . . . . 8 (𝐴 = ∅ → (1o𝐴) = (1o ∖ ∅))
3 dif0 3508 . . . . . . . 8 (1o ∖ ∅) = 1o
42, 3eqtrdi 2238 . . . . . . 7 (𝐴 = ∅ → (1o𝐴) = 1o)
5 difeq2 3262 . . . . . . . 8 (𝐴 = 1o → (1o𝐴) = (1o ∖ 1o))
6 difid 3506 . . . . . . . 8 (1o ∖ 1o) = ∅
75, 6eqtrdi 2238 . . . . . . 7 (𝐴 = 1o → (1o𝐴) = ∅)
84, 7orim12i 760 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = 1o ∨ (1o𝐴) = ∅))
98orcomd 730 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
101, 9syl 14 . . . 4 (𝐴 ∈ {∅, 1o} → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
11 1on 6449 . . . . . 6 1o ∈ On
12 difexg 4159 . . . . . 6 (1o ∈ On → (1o𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1o𝐴) ∈ V
1413elpr 3628 . . . 4 ((1o𝐴) ∈ {∅, 1o} ↔ ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
1510, 14sylibr 134 . . 3 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ {∅, 1o})
16 df2o3 6456 . . 3 2o = {∅, 1o}
1715, 16eleqtrrdi 2283 . 2 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ 2o)
1817, 16eleq2s 2284 1 (𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709   = wceq 1364  wcel 2160  Vcvv 2752  cdif 3141  c0 3437  {cpr 3608  Oncon0 4381  1oc1o 6435  2oc2o 6436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-tr 4117  df-iord 4384  df-on 4386  df-suc 4389  df-1o 6442  df-2o 6443
This theorem is referenced by:  ismkvnex  7184
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