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Theorem psrel 17184
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 17183 . . 3 (𝐴 ∈ PosetRel → (𝐴 ∈ PosetRel ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴))))
21ibi 256 . 2 (𝐴 ∈ PosetRel → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ 𝐴 ∧ (𝐴𝐴) = ( I ↾ 𝐴)))
32simp1d 1071 1 (𝐴 ∈ PosetRel → Rel 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  cin 3566  wss 3567   cuni 4427   I cid 5013  ccnv 5103  cres 5106  ccom 5108  Rel wrel 5109  PosetRelcps 17179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-in 3574  df-ss 3581  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-res 5116  df-ps 17181
This theorem is referenced by:  pslem  17187  cnvps  17193  psss  17195  cnvtsr  17203  tsrdir  17219
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