Theorem psref 18728

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 18727	. . . . . 6 ⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
3	2	simpld 500	. . . . 5 ⊢ (𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅)
4	1, 3	eqtrid 2808	. . . 4 ⊢ (𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅)
5	4	eleq2d 2847	. . 3 ⊢ (𝑅 ∈ PosetRel → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅))
6		pslem 18726	. . . 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 412	1 ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∪ cuni 4867   class class class wbr 5103  dom cdm 5651  ran crn 5652  PosetRelcps 18718

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ps 18720

This theorem is used by:  psss  18734  psssdm2  18735  ordtt1  23677