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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 17814 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) | |
2 | 1 | ibi 269 | . 2 ⊢ (𝑅 ∈ PosetRel → (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))) |
3 | 2 | simp2d 1139 | 1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 ∪ cuni 4840 I cid 5461 ◡ccnv 5556 ↾ cres 5559 ∘ ccom 5561 Rel wrel 5562 PosetRelcps 17810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-in 3945 df-ss 3954 df-uni 4841 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-res 5569 df-ps 17812 |
This theorem is referenced by: pslem 17818 cnvps 17824 psss 17826 tsrdir 17850 |
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