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Theorem psrel 18614
Description: A poset is a relation. (Contributed by NM, 12-May-2008.)
Assertion
Ref Expression
psrel (𝐴 ∈ PosetRel → Rel 𝐴)

Proof of Theorem psrel
StepHypRef Expression
1 isps 18613 . . 3 (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴))))
21ibi 267 . 2 (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴)))
32simp1d 1143 1 (𝐴 ∈ PosetRel → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951   cuni 4907   I cid 5577  ccnv 5684  cres 5687  ccom 5689  Rel wrel 5690  PosetRelcps 18609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-res 5697  df-ps 18611
This theorem is referenced by:  pslem  18617  cnvps  18623  psss  18625  cnvtsr  18633  tsrdir  18649
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