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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 6331 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
2 | ordtr 6331 | . . 3 ⊢ (Ord 𝐵 → Tr 𝐵) | |
3 | trin 5234 | . . 3 ⊢ ((Tr 𝐴 ∧ Tr 𝐵) → Tr (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Tr (𝐴 ∩ 𝐵)) |
5 | inss2 4189 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
6 | trssord 6334 | . . 3 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐵 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
7 | 5, 6 | mp3an2 1449 | . 2 ⊢ ((Tr (𝐴 ∩ 𝐵) ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
8 | 4, 7 | sylancom 588 | 1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∩ cin 3909 ⊆ wss 3910 Tr wtr 5222 Ord word 6316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rab 3408 df-v 3447 df-in 3917 df-ss 3927 df-uni 4866 df-tr 5223 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-ord 6320 |
This theorem is referenced by: onin 6348 ordtri3or 6349 ordelinel 6418 smores 8298 smores2 8300 ordtypelem5 9458 ordtypelem7 9460 |
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