Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6207 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | ordtr 6207 | . . 3 ⊢ (Ord 𝐵 → Tr 𝐵) | |
3 | trin 5184 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵)) |
5 | inss2 4208 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
6 | trssord 6210 | . . 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 3937 ⊆ wss 3938 Tr wtr 5174 Ord word 6192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-in 3945 df-ss 3954 df-uni 4841 df-tr 5175 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 |
This theorem is referenced by: onin 6224 ordtri3or 6225 ordelinel 6291 smores 7991 smores2 7993 ordtypelem5 8988 ordtypelem7 8990 |
Copyright terms: Public domain | W3C validator |