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Mirrors > Home > MPE Home > Th. List > psrel | Structured version Visualization version GIF version |
Description: A poset is a relation. (Contributed by NM, 12-May-2008.) |
Ref | Expression |
---|---|
psrel | ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isps 17800 | . . 3 ⊢ (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴)))) | |
2 | 1 | ibi 268 | . 2 ⊢ (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴 ∘ 𝐴) ⊆ 𝐴 ∧ (𝐴 ∩ ◡𝐴) = ( I ↾ ∪ ∪ 𝐴))) |
3 | 2 | simp1d 1134 | 1 ⊢ (𝐴 ∈ PosetRel → Rel 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∪ cuni 4830 I cid 5452 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 PosetRelcps 17796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-in 3940 df-ss 3949 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-res 5560 df-ps 17798 |
This theorem is referenced by: pslem 17804 cnvps 17810 psss 17812 cnvtsr 17820 tsrdir 17836 |
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