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) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))

Proof of Theorem ordelinel

Step	Hyp	Ref	Expression
1		ordtri2or3 6464	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
2	1	3adant3 1129	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
3		eleq1a 2820	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 = (𝐴 ∩ 𝐵) → 𝐴 ∈ 𝐶))
4		eleq1a 2820	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐵 = (𝐴 ∩ 𝐵) → 𝐵 ∈ 𝐶))
5	3, 4	orim12d 962	. . 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 6394	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
8		inss1 4223	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
9		ordtr2 6408	. . . . 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 4224	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
12		ordtr2 6408	. . . . 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 857	. . 3 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
15	7, 14	stoic3 1770	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
16	6, 15	impbid 211	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))

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)