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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 3482 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | inex1 5323 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
4 | id 22 | . . . 4 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → 𝑧 = (𝑥 ∩ 𝑦)) | |
5 | 4, 4 | coeq12d 5878 | . . 3 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ∘ 𝑧) = ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦))) |
6 | 5, 4 | sseq12d 4029 | . 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 5878 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∘ 𝑧) = (𝑥 ∘ 𝑥)) |
9 | 8, 7 | sseq12d 4029 | . 2 ⊢ (𝑧 = 𝑥 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑥 ∘ 𝑥) ⊆ 𝑥)) |
10 | id 22 | . . . 4 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) | |
11 | 10, 10 | coeq12d 5878 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∘ 𝑧) = (𝑦 ∘ 𝑦)) |
12 | 11, 10 | sseq12d 4029 | . 2 ⊢ (𝑧 = 𝑦 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑦 ∘ 𝑦) ⊆ 𝑦)) |
13 | trin2 6146 | . 2 ⊢ (((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦) → ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦)) | |
14 | 1, 3, 6, 9, 12, 13 | cllem0 43556 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 {cab 2712 ∀wral 3059 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∘ ccom 5693 |
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-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-co 5698 |
This theorem is referenced by: (None) |
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