Theorem psref 18619

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 ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Proof of Theorem psref

Step	Hyp	Ref	Expression
1		psref.1	. . . . 5 ⊢ 𝑋 = dom 𝑅
2		psdmrn 18618	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3	2	simpld 494	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
4	1, 3	eqtrid 2789	. . . 4 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅)
5	4	eleq2d 2827	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
6		pslem 18617	. . . 4 ⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴 ∧ 𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴 ∧ 𝐴𝑅𝐴) → 𝐴 = 𝐴)))
7	6	simp2d 1144	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
8	5, 7	sylbid 240	. 2 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 → 𝐴𝑅𝐴))
9	8	imp 406	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∪ cuni 4907   class class class wbr 5143  dom cdm 5685  ran crn 5686  PosetRelcps 18609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ps 18611

This theorem is referenced by:  psss  18625  psssdm2  18626  ordtt1  23387