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Theorem psref 18569
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 18568 . . . . . 6 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
32simpld 493 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
41, 3eqtrid 2777 . . . 4 (𝑅 ∈ PosetRel → 𝑋 = 𝑅)
54eleq2d 2811 . . 3 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴 𝑅))
6 pslem 18567 . . . 4 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴 = 𝐴)))
76simp2d 1140 . . 3 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
85, 7sylbid 239 . 2 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴𝑅𝐴))
98imp 405 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098   cuni 4909   class class class wbr 5149  dom cdm 5678  ran crn 5679  PosetRelcps 18559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ps 18561
This theorem is referenced by:  psss  18575  psssdm2  18576  ordtt1  23327
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