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Mirrors > Home > MPE Home > Th. List > Mathboxes > trficl | Structured version Visualization version GIF version |
Description: The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.) |
Ref | Expression |
---|---|
trficl.a | ⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧} |
Ref | Expression |
---|---|
trficl | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trficl.a | . 2 ⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧} | |
2 | vex 3465 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | inex1 5318 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
4 | id 22 | . . . 4 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → 𝑧 = (𝑥 ∩ 𝑦)) | |
5 | 4, 4 | coeq12d 5867 | . . 3 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ∘ 𝑧) = ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦))) |
6 | 5, 4 | sseq12d 4010 | . 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦))) |
7 | id 22 | . . . 4 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥) | |
8 | 7, 7 | coeq12d 5867 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∘ 𝑧) = (𝑥 ∘ 𝑥)) |
9 | 8, 7 | sseq12d 4010 | . 2 ⊢ (𝑧 = 𝑥 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑥 ∘ 𝑥) ⊆ 𝑥)) |
10 | id 22 | . . . 4 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) | |
11 | 10, 10 | coeq12d 5867 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∘ 𝑧) = (𝑦 ∘ 𝑦)) |
12 | 11, 10 | sseq12d 4010 | . 2 ⊢ (𝑧 = 𝑦 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑦 ∘ 𝑦) ⊆ 𝑦)) |
13 | trin2 6130 | . 2 ⊢ (((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦) → ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦)) | |
14 | 1, 3, 6, 9, 12, 13 | cllem0 43138 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {cab 2702 ∀wral 3050 Vcvv 3461 ∩ cin 3943 ⊆ wss 3944 ∘ ccom 5682 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-co 5687 |
This theorem is referenced by: (None) |
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