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Mirrors > Home > MPE Home > Th. List > Mathboxes > ontopbas | Structured version Visualization version GIF version |
Description: An ordinal number is a topological basis. (Contributed by Chen-Pang He, 8-Oct-2015.) |
Ref | Expression |
---|---|
ontopbas | ⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onelon 6411 | . . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On) | |
2 | onelon 6411 | . . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ On) | |
3 | 1, 2 | anim12dan 619 | . . . . . . 7 ⊢ ((𝐵 ∈ On ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ On ∧ 𝑦 ∈ On)) |
4 | 3 | ex 412 | . . . . . 6 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ On ∧ 𝑦 ∈ On))) |
5 | onin 6417 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∩ 𝑦) ∈ On) | |
6 | 4, 5 | syl6 35 | . . . . 5 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∩ 𝑦) ∈ On)) |
7 | 6 | anc2ri 556 | . . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ∈ On ∧ 𝐵 ∈ On))) |
8 | inss1 4245 | . . . . . . 7 ⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥 | |
9 | 8 | jctl 523 | . . . . . 6 ⊢ (𝑥 ∈ 𝐵 → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵))) |
12 | ontr2 6433 | . . . 4 ⊢ (((𝑥 ∩ 𝑦) ∈ On ∧ 𝐵 ∈ On) → (((𝑥 ∩ 𝑦) ⊆ 𝑥 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐵)) | |
13 | 7, 11, 12 | syl6c 70 | . . 3 ⊢ (𝐵 ∈ On → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∩ 𝑦) ∈ 𝐵)) |
14 | 13 | ralrimivv 3198 | . 2 ⊢ (𝐵 ∈ On → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) |
15 | fiinbas 22975 | . 2 ⊢ ((𝐵 ∈ On ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∩ 𝑦) ∈ 𝐵) → 𝐵 ∈ TopBases) | |
16 | 14, 15 | mpdan 687 | 1 ⊢ (𝐵 ∈ On → 𝐵 ∈ TopBases) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∀wral 3059 ∩ cin 3962 ⊆ wss 3963 Oncon0 6386 TopBasesctb 22968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-bases 22969 |
This theorem is referenced by: onsstopbas 36412 onsuctop 36416 |
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