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Theorem psref 17594
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 17593 . . . . . 6 (𝑅 ∈ PosetRel → (dom 𝑅 = 𝑅 ∧ ran 𝑅 = 𝑅))
32simpld 490 . . . . 5 (𝑅 ∈ PosetRel → dom 𝑅 = 𝑅)
41, 3syl5eq 2826 . . . 4 (𝑅 ∈ PosetRel → 𝑋 = 𝑅)
54eleq2d 2845 . . 3 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴 𝑅))
6 pslem 17592 . . . 4 (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 𝑅𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴𝐴𝑅𝐴) → 𝐴 = 𝐴)))
76simp2d 1134 . . 3 (𝑅 ∈ PosetRel → (𝐴 𝑅𝐴𝑅𝐴))
85, 7sylbid 232 . 2 (𝑅 ∈ PosetRel → (𝐴𝑋𝐴𝑅𝐴))
98imp 397 1 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107   cuni 4671   class class class wbr 4886  dom cdm 5355  ran crn 5356  PosetRelcps 17584
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-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
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-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ps 17586
This theorem is referenced by:  psss  17600  psssdm2  17601  ordtt1  21591
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