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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 17928 | . . 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 | simp3d 1145 | 1 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 ⊆ wss 3843 ∪ cuni 4796 I cid 5428 ◡ccnv 5524 ↾ cres 5527 ∘ ccom 5529 Rel wrel 5530 PosetRelcps 17924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3400 df-in 3850 df-ss 3860 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-res 5537 df-ps 17926 |
This theorem is referenced by: pslem 17932 cnvps 17938 tsrdir 17964 |
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