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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 3497 | . . 3 ⊢ 𝑥 ∈ V | |
3 | 2 | inex1 5213 | . 2 ⊢ (𝑥 ∩ 𝑦) ∈ V |
4 | id 22 | . . . 4 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → 𝑧 = (𝑥 ∩ 𝑦)) | |
5 | 4, 4 | coeq12d 5729 | . . 3 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ∘ 𝑧) = ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦))) |
6 | 5, 4 | sseq12d 3999 | . 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 5729 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∘ 𝑧) = (𝑥 ∘ 𝑥)) |
9 | 8, 7 | sseq12d 3999 | . 2 ⊢ (𝑧 = 𝑥 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑥 ∘ 𝑥) ⊆ 𝑥)) |
10 | id 22 | . . . 4 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦) | |
11 | 10, 10 | coeq12d 5729 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∘ 𝑧) = (𝑦 ∘ 𝑦)) |
12 | 11, 10 | sseq12d 3999 | . 2 ⊢ (𝑧 = 𝑦 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑦 ∘ 𝑦) ⊆ 𝑦)) |
13 | trin2 5977 | . 2 ⊢ (((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦) → ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦)) | |
14 | 1, 3, 6, 9, 12, 13 | cllem0 39918 | 1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∘ ccom 5553 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-co 5558 |
This theorem is referenced by: (None) |
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