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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 18456 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) | |
2 | 1 | ibi 266 | . 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))) |
3 | 2 | simp2d 1143 | 1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 ⊆ wss 3910 ∪ cuni 4865 I cid 5530 ◡ccnv 5632 ↾ cres 5635 ∘ ccom 5637 Rel wrel 5638 PosetRelcps 18452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3408 df-v 3447 df-in 3917 df-ss 3927 df-uni 4866 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-res 5645 df-ps 18454 |
This theorem is referenced by: pslem 18460 cnvps 18466 psss 18468 tsrdir 18492 |
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