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Theorem ordelinel 5789
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 5788 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
213adant3 1079 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
3 eleq1a 2693 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐴 = (𝐴𝐵) → 𝐴𝐶))
4 eleq1a 2693 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐵 = (𝐴𝐵) → 𝐵𝐶))
53, 4orim12d 882 . . 3 ((𝐴𝐵) ∈ 𝐶 → ((𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)) → (𝐴𝐶𝐵𝐶)))
62, 5syl5com 31 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
7 ordin 5717 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
8 inss1 3816 . . . . 5 (𝐴𝐵) ⊆ 𝐴
9 ordtr2 5732 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐴𝐴𝐶) → (𝐴𝐵) ∈ 𝐶))
108, 9mpani 711 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐴𝐶 → (𝐴𝐵) ∈ 𝐶))
11 inss2 3817 . . . . 5 (𝐴𝐵) ⊆ 𝐵
12 ordtr2 5732 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐵𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
1311, 12mpani 711 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐵𝐶 → (𝐴𝐵) ∈ 𝐶))
1410, 13jaod 395 . . 3 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
157, 14stoic3 1698 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
166, 15impbid 202 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  cin 3558  wss 3559  Ord word 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-ord 5690
This theorem is referenced by:  mreexexdOLD  16241
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