Theorem psref 18540

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 18539	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3	2	simpld 494	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
4	1, 3	eqtrid 2783	. . . 4 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅)
5	4	eleq2d 2822	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
6		pslem 18538	. . . 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 1542   ∈ wcel 2114  ∪ cuni 4850   class class class wbr 5085  dom cdm 5631  ran crn 5632  PosetRelcps 18530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ps 18532

This theorem is referenced by:  psss  18546  psssdm2  18547  ordtt1  23344