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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 6291 | . . . 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 2911 | . . . 4 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐶 → (𝐴 = (𝐴 ∩ 𝐵) → 𝐴 ∈ 𝐶)) | |
4 | eleq1a 2911 | . . . 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 6224 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
8 | inss1 4208 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
9 | ordtr2 6238 | . . . . 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 4209 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
12 | ordtr2 6238 | . . . . 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 1776 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∩ cin 3938 ⊆ wss 3939 Ord word 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 |
This theorem is referenced by: (None) |
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