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Theorem psref 17031
 Description: A poset is reflexive. (Contributed by NM, 13-May-2008.)
Hypothesis
Ref Expression
psref.1 𝑋 = dom 𝑅
Assertion
Ref Expression
psref ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)

Proof of Theorem psref
StepHypRef Expression
1 psref.1 . . . . 5 𝑋 = dom 𝑅
2 psdmrn 17030 . . . . . 6 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
32simpld 474 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
41, 3syl5eq 2656 . . . 4 (𝑅 ∈ PosetRel → 𝑋 = 𝑅)
54eleq2d 2673 . . 3 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴 𝑅))
6 pslem 17029 . . . 4 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴 = 𝐴)))
76simp2d 1067 . . 3 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
85, 7sylbid 229 . 2 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴𝑅𝐴))
98imp 444 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∪ cuni 4372   class class class wbr 4583  dom cdm 5038  ran crn 5039  PosetRelcps 17021 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ps 17023 This theorem is referenced by:  psss  17037  psssdm2  17038  ordtt1  20993
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