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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 6205 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | ordtr 6205 | . . 3 ⊢ (Ord 𝐵 → Tr 𝐵) | |
3 | trin 5156 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 599 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵)) |
5 | inss2 4130 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
6 | trssord 6208 | . . 3 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | mp3an2 1451 | . 2 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
8 | 4, 7 | sylancom 591 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∩ cin 3852 ⊆ wss 3853 Tr wtr 5146 Ord word 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-in 3860 df-ss 3870 df-uni 4806 df-tr 5147 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 |
This theorem is referenced by: onin 6222 ordtri3or 6223 ordelinel 6289 smores 8067 smores2 8069 ordtypelem5 9116 ordtypelem7 9118 |
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