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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 6394 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 2 | eloni 6394 | . . 3 ⊢ (𝐵 ∈ On → Ord 𝐵) | |
| 3 | ordin 6414 | . . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵)) | |
| 4 | 1, 2, 3 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵)) |
| 5 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On) | |
| 6 | inex1g 5319 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V) | |
| 7 | elong 6392 | . . 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 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-in 3958 df-ss 3968 df-uni 4908 df-tr 5260 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: tfrlem5 8420 noreson 27705 ontopbas 36429 |
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