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Theorem 2oconcl 6343
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 3554 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
2 difeq2 3192 . . . . . . . 8 (𝐴 = ∅ → (1o𝐴) = (1o ∖ ∅))
3 dif0 3437 . . . . . . . 8 (1o ∖ ∅) = 1o
42, 3eqtrdi 2189 . . . . . . 7 (𝐴 = ∅ → (1o𝐴) = 1o)
5 difeq2 3192 . . . . . . . 8 (𝐴 = 1o → (1o𝐴) = (1o ∖ 1o))
6 difid 3435 . . . . . . . 8 (1o ∖ 1o) = ∅
75, 6eqtrdi 2189 . . . . . . 7 (𝐴 = 1o → (1o𝐴) = ∅)
84, 7orim12i 749 . . . . . 6 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = 1o ∨ (1o𝐴) = ∅))
98orcomd 719 . . . . 5 ((𝐴 = ∅ ∨ 𝐴 = 1o) → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
101, 9syl 14 . . . 4 (𝐴 ∈ {∅, 1o} → ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
11 1on 6327 . . . . . 6 1o ∈ On
12 difexg 4076 . . . . . 6 (1o ∈ On → (1o𝐴) ∈ V)
1311, 12ax-mp 5 . . . . 5 (1o𝐴) ∈ V
1413elpr 3552 . . . 4 ((1o𝐴) ∈ {∅, 1o} ↔ ((1o𝐴) = ∅ ∨ (1o𝐴) = 1o))
1510, 14sylibr 133 . . 3 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ {∅, 1o})
16 df2o3 6334 . . 3 2o = {∅, 1o}
1715, 16eleqtrrdi 2234 . 2 (𝐴 ∈ {∅, 1o} → (1o𝐴) ∈ 2o)
1817, 16eleq2s 2235 1 (𝐴 ∈ 2o → (1o𝐴) ∈ 2o)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 698   = wceq 1332  wcel 1481  Vcvv 2689  cdif 3072  c0 3367  {cpr 3532  Oncon0 4292  1oc1o 6313  2oc2o 6314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-uni 3744  df-tr 4034  df-iord 4295  df-on 4297  df-suc 4300  df-1o 6320  df-2o 6321
This theorem is referenced by:  ismkvnex  7036
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