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Mirrors > Home > MPE Home > Th. List > psref2 | Structured version Visualization version GIF version |
Description: A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.) |
Ref | Expression |
---|---|
psref2 | ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isps 17811 | . . 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 | simp3d 1140 | 1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∩ cin 3934 ⊆ wss 3935 ∪ cuni 4837 I cid 5458 ◡ccnv 5553 ↾ cres 5556 ∘ ccom 5558 Rel wrel 5559 PosetRelcps 17807 |
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 |
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-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-res 5566 df-ps 17809 |
This theorem is referenced by: pslem 17815 cnvps 17821 tsrdir 17847 |
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