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Mirrors > Home > MPE Home > Th. List > pstr2 | Structured version Visualization version GIF version |
Description: A poset is transitive. (Contributed by FL, 3-Aug-2009.) |
Ref | Expression |
---|---|
pstr2 | ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isps 18028 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) | |
2 | 1 | ibi 270 | . 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))) |
3 | 2 | simp2d 1145 | 1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∩ cin 3852 ⊆ wss 3853 ∪ cuni 4805 I cid 5439 ◡ccnv 5535 ↾ cres 5538 ∘ ccom 5540 Rel wrel 5541 PosetRelcps 18024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-in 3860 df-ss 3870 df-uni 4806 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-res 5548 df-ps 18026 |
This theorem is referenced by: pslem 18032 cnvps 18038 psss 18040 tsrdir 18064 |
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