Theorem psref

Description: A poset is reflexive. (Contributed by NM, 13-May-2008)

	Ref	Expression
Hypothesis	psref.1	⊢ 𝑋 = dom 𝑅
Assertion	psref	⊢ ( ( 𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋 ) → 𝐴 𝑅 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		psref.1	⊢ 𝑋 = dom 𝑅
2		psdmrn	⊢ ( 𝑅 ∈ PosetRel → ( dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅 ) )
3	2	simpld	⊢ ( 𝑅 ∈ PosetRel → dom 𝑅 = ∪ ∪ 𝑅 )
4	1 3	eqtrid	⊢ ( 𝑅 ∈ PosetRel → 𝑋 = ∪ ∪ 𝑅 )
5	4	eleq2d	⊢ ( 𝑅 ∈ PosetRel → ( 𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ∪ ∪ 𝑅 ) )
6		pslem	⊢ ( 𝑅 ∈ PosetRel → ( ( ( 𝐴 𝑅 𝐴 ∧ 𝐴 𝑅 𝐴 ) → 𝐴 𝑅 𝐴 ) ∧ ( 𝐴 ∈ ∪ ∪ 𝑅 → 𝐴 𝑅 𝐴 ) ∧ ( ( 𝐴 𝑅 𝐴 ∧ 𝐴 𝑅 𝐴 ) → 𝐴 = 𝐴 ) ) )
7	6	simp2d	⊢ ( 𝑅 ∈ PosetRel → ( 𝐴 ∈ ∪ ∪ 𝑅 → 𝐴 𝑅 𝐴 ) )
8	5 7	sylbid	⊢ ( 𝑅 ∈ PosetRel → ( 𝐴 ∈ 𝑋 → 𝐴 𝑅 𝐴 ) )
9	8	imp	⊢ ( ( 𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋 ) → 𝐴 𝑅 𝐴 )