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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 5650 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 5650 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordin 5670 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 493 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵)) |
5 | simpl 472 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | |
6 | inex1g 4729 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V) | |
7 | elong 5648 | . . 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 246 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 Ord word 5639 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-tr 4681 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: tfrlem5 7363 noreson 31057 ontopbas 31597 |
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