MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordelinel Structured version   Visualization version   GIF version

Theorem ordelinel 6487
Description: The intersection of two ordinal classes is an element of a third if and only if either one of them is. (Contributed by David Moews, 1-May-2017.) (Proof shortened by JJ, 24-Sep-2021.)
Assertion
Ref Expression
ordelinel ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Proof of Theorem ordelinel
StepHypRef Expression
1 ordtri2or3 6486 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
213adant3 1131 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
3 eleq1a 2834 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐴 = (𝐴𝐵) → 𝐴𝐶))
4 eleq1a 2834 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐵 = (𝐴𝐵) → 𝐵𝐶))
53, 4orim12d 966 . . 3 ((𝐴𝐵) ∈ 𝐶 → ((𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)) → (𝐴𝐶𝐵𝐶)))
62, 5syl5com 31 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
7 ordin 6416 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
8 inss1 4245 . . . . 5 (𝐴𝐵) ⊆ 𝐴
9 ordtr2 6430 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐴𝐴𝐶) → (𝐴𝐵) ∈ 𝐶))
108, 9mpani 696 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐴𝐶 → (𝐴𝐵) ∈ 𝐶))
11 inss2 4246 . . . . 5 (𝐴𝐵) ⊆ 𝐵
12 ordtr2 6430 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐵𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
1311, 12mpani 696 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐵𝐶 → (𝐴𝐵) ∈ 𝐶))
1410, 13jaod 859 . . 3 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
157, 14stoic3 1773 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
166, 15impbid 212 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963  Ord word 6385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator