Theorem ordelinelOLD 5826

Description: Obsolete proof of ordelinel 5825 as of 24-Sep-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
ordelinelOLD	⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))

Proof of Theorem ordelinelOLD

Step	Hyp	Ref	Expression
1		ordtri2or3 5824	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
2	1	3adant3 1081	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)))
3		eleq1 2689	. . . . 5 ⊢ (𝐴 = (𝐴 ∩ 𝐵) → (𝐴 ∈ 𝐶 ↔ (𝐴 ∩ 𝐵) ∈ 𝐶))
4		orc 400	. . . . 5 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶))
5	3, 4	syl6bir 244	. . . 4 ⊢ (𝐴 = (𝐴 ∩ 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
6		eleq1 2689	. . . . 5 ⊢ (𝐵 = (𝐴 ∩ 𝐵) → (𝐵 ∈ 𝐶 ↔ (𝐴 ∩ 𝐵) ∈ 𝐶))
7		olc 399	. . . . 5 ⊢ (𝐵 ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶))
8	6, 7	syl6bir 244	. . . 4 ⊢ (𝐵 = (𝐴 ∩ 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
9	5, 8	jaoi 394	. . 3 ⊢ ((𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵)) → ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
10	2, 9	syl 17	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))
11		inss1 3833	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
12		ordin 5753	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
13		ordtr2 5768	. . . . 5 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
14	12, 13	stoic3 1701	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
15	11, 14	mpani 712	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ∈ 𝐶 → (𝐴 ∩ 𝐵) ∈ 𝐶))
16		inss2 3834	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
17		ordtr2 5768	. . . . 5 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
18	12, 17	stoic3 1701	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
19	16, 18	mpani 712	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 → (𝐴 ∩ 𝐵) ∈ 𝐶))
20	15, 19	jaod 395	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶))
21	10, 20	impbid 202	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574  Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by: (None)