Theorem psref 18631

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 18630	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3	2	simpld 499	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
4	1, 3	eqtrid 2810	. . . 4 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅)
5	4	eleq2d 2849	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
6		pslem 18629	. . . 4 ⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐴 ∧ 𝐴𝑅𝐴) → 𝐴𝑅𝐴) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐴 ∧ 𝐴𝑅𝐴) → 𝐴 = 𝐴)))
7	6	simp2d 1161	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴))
8	5, 7	sylbid 243	. 2 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 → 𝐴𝑅𝐴))
9	8	imp 411	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∪ cuni 4873   class class class wbr 5110  dom cdm 5663  ran crn 5664  PosetRelcps 18621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ps 18623

This theorem is referenced by:  psss  18637  psssdm2  18638  ordtt1  23517