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Theorem ordelinel 6465
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 6464 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
213adant3 1129 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
3 eleq1a 2820 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐴 = (𝐴𝐵) → 𝐴𝐶))
4 eleq1a 2820 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐵 = (𝐴𝐵) → 𝐵𝐶))
53, 4orim12d 962 . . 3 ((𝐴𝐵) ∈ 𝐶 → ((𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)) → (𝐴𝐶𝐵𝐶)))
62, 5syl5com 31 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
7 ordin 6394 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
8 inss1 4223 . . . . 5 (𝐴𝐵) ⊆ 𝐴
9 ordtr2 6408 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐴𝐴𝐶) → (𝐴𝐵) ∈ 𝐶))
108, 9mpani 694 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐴𝐶 → (𝐴𝐵) ∈ 𝐶))
11 inss2 4224 . . . . 5 (𝐴𝐵) ⊆ 𝐵
12 ordtr2 6408 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐵𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
1311, 12mpani 694 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐵𝐶 → (𝐴𝐵) ∈ 𝐶))
1410, 13jaod 857 . . 3 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
157, 14stoic3 1770 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
166, 15impbid 211 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  cin 3938  wss 3939  Ord word 6363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6367
This theorem is referenced by: (None)
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