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Theorem psrel 17801
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 17800 . . 3 (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴))))
21ibi 268 . 2 (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴)))
32simp1d 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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