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Mirrors > Home > MPE Home > Th. List > ordelinel | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ordelinel | ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtri2or3 5741 | . . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) | |
2 | 1 | 3adant3 1074 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴 ∩ 𝐵) ∨ 𝐵 = (𝐴 ∩ 𝐵))) |
3 | eleq1a 2683 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 = (𝐴 ∩ 𝐵) → 𝐴 ∈ 𝐶)) | |
4 | eleq1a 2683 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐵 = (𝐴 ∩ 𝐵) → 𝐵 ∈ 𝐶)) | |
5 | 3, 4 | orim12d 879 | . . 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 5670 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
8 | inss1 3795 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
9 | ordtr2 5685 | . . . . 5 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)) | |
10 | 8, 9 | mpani 708 | . . . 4 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (𝐴 ∈ 𝐶 → (𝐴 ∩ 𝐵) ∈ 𝐶)) |
11 | inss2 3796 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
12 | ordtr2 5685 | . . . . 5 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (((𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)) | |
13 | 11, 12 | mpani 708 | . . . 4 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → (𝐵 ∈ 𝐶 → (𝐴 ∩ 𝐵) ∈ 𝐶)) |
14 | 10, 13 | jaod 394 | . . 3 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)) |
15 | 7, 14 | stoic3 1692 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶)) |
16 | 6, 15 | impbid 201 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴 ∩ 𝐵) ∈ 𝐶 ↔ (𝐴 ∈ 𝐶 ∨ 𝐵 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 Ord word 5639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 |
This theorem is referenced by: mreexexdOLD 16132 |
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