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Mirrors > Home > MPE Home > Th. List > onin | Structured version Visualization version GIF version |
Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.) |
Ref | Expression |
---|---|
onin | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6365 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 6365 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordin 6385 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵)) |
5 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | |
6 | inex1g 5310 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V) | |
7 | elong 6363 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → ((𝐴 ∩ 𝐵) ∈ On ↔ Ord (𝐴 ∩ 𝐵))) | |
8 | 5, 6, 7 | 3syl 18 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∩ 𝐵) ∈ On ↔ Ord (𝐴 ∩ 𝐵))) |
9 | 4, 8 | mpbird 257 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2098 Vcvv 3466 ∩ cin 3940 Ord word 6354 Oncon0 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-in 3948 df-ss 3958 df-uni 4901 df-tr 5257 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-ord 6358 df-on 6359 |
This theorem is referenced by: tfrlem5 8376 noreson 27512 ontopbas 35804 |
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