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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 6201 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
2 | eloni 6201 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
3 | ordin 6221 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
4 | 1, 2, 3 | syl2an 597 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵)) |
5 | simpl 485 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | |
6 | inex1g 5223 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V) | |
7 | elong 6199 | . . 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 259 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 Ord word 6190 Oncon0 6191 |
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 2793 ax-sep 5203 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-tr 5173 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 |
This theorem is referenced by: tfrlem5 8016 noreson 33167 ontopbas 33776 |
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