Theorem ordelinel 6283

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) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))

Proof of Theorem ordelinel

Step	Hyp	Ref	Expression
1		ordtri2or3 6282	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
2	1	3adant3 1128	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
3		eleq1a 2908	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 = (𝐴 ∩ 𝐵) → 𝐴 ∈ 𝐶))
4		eleq1a 2908	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐵 = (𝐴 ∩ 𝐵) → 𝐵 ∈ 𝐶))
5	3, 4	orim12d 961	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → ((𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)) → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
6	2, 5	syl5com 31	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
7		ordin 6215	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
8		inss1 4204	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
9		ordtr2 6229	. . . . 5 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
10	8, 9	mpani 694	. . . 4 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (𝐴 ∈ 𝐶 → (𝐴 ∩ 𝐵) ∈ 𝐶))
11		inss2 4205	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
12		ordtr2 6229	. . . . 5 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
13	11, 12	mpani 694	. . . 4 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 → (𝐴 ∩ 𝐵) ∈ 𝐶))
14	10, 13	jaod 855	. . 3 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
15	7, 14	stoic3 1773	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
16	6, 15	impbid 214	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ∩ cin 3934   ⊆ wss 3935  Ord word 6184

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-tr 5165  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-ord 6188

This theorem is referenced by: (None)