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Mirrors > Home > MPE Home > Th. List > ordin | Structured version Visualization version GIF version |
Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.) |
Ref | Expression |
---|---|
ordin | ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtr 5654 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | ordtr 5654 | . . 3 ⊢ (Ord 𝐵 → Tr 𝐵) | |
3 | trin 4691 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 493 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵)) |
5 | inss2 3796 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
6 | trssord 5657 | . . 3 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | mp3an2 1404 | . 2 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
8 | 4, 7 | sylancom 698 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∩ cin 3539 ⊆ wss 3540 Tr wtr 4680 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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 |
This theorem is referenced by: onin 5671 ordtri3or 5672 ordelinel 5742 ordelinelOLD 5743 smores 7336 smores2 7338 ordtypelem5 8310 ordtypelem7 8312 |
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