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Theorem psrel 17593
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 17592 . . 3 (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴))))
21ibi 259 . 2 (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴)))
32simp1d 1133 1 (𝐴 ∈ PosetRel → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1601  wcel 2107  cin 3791  wss 3792   cuni 4673   I cid 5262  ccnv 5356  cres 5359  ccom 5361  Rel wrel 5362  PosetRelcps 17588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rex 3096  df-v 3400  df-in 3799  df-ss 3806  df-uni 4674  df-br 4889  df-opab 4951  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-res 5369  df-ps 17590
This theorem is referenced by:  pslem  17596  cnvps  17602  psss  17604  cnvtsr  17612  tsrdir  17628
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