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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 6208 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | ordtr 6208 | . . 3 ⊢ (Ord 𝐵 → Tr 𝐵) | |
3 | trin 5185 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵)) |
5 | inss2 4209 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
6 | trssord 6211 | . . 3 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | mp3an2 1445 | . 2 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
8 | 4, 7 | sylancom 590 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∩ cin 3938 ⊆ wss 3939 Tr wtr 5175 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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-in 3946 df-ss 3955 df-uni 4842 df-tr 5176 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 |
This theorem is referenced by: onin 6225 ordtri3or 6226 ordelinel 6292 smores 7992 smores2 7994 ordtypelem5 8989 ordtypelem7 8991 |
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