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Theorem ordelinel 5967
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 5966 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
213adant3 1126 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
3 eleq1a 2845 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐴 = (𝐴𝐵) → 𝐴𝐶))
4 eleq1a 2845 . . . 4 ((𝐴𝐵) ∈ 𝐶 → (𝐵 = (𝐴𝐵) → 𝐵𝐶))
53, 4orim12d 949 . . 3 ((𝐴𝐵) ∈ 𝐶 → ((𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)) → (𝐴𝐶𝐵𝐶)))
62, 5syl5com 31 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
7 ordin 5895 . . 3 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
8 inss1 3981 . . . . 5 (𝐴𝐵) ⊆ 𝐴
9 ordtr2 5910 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐴𝐴𝐶) → (𝐴𝐵) ∈ 𝐶))
108, 9mpani 676 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐴𝐶 → (𝐴𝐵) ∈ 𝐶))
11 inss2 3982 . . . . 5 (𝐴𝐵) ⊆ 𝐵
12 ordtr2 5910 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐵𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
1311, 12mpani 676 . . . 4 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (𝐵𝐶 → (𝐴𝐵) ∈ 𝐶))
1410, 13jaod 848 . . 3 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
157, 14stoic3 1849 . 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  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  cin 3722  wss 3723  Ord word 5864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868
This theorem is referenced by: (None)
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