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Metamath Proof Explorer

Theorem flddivrng

Description: A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010) (Revised by Mario Carneiro, 15-Dec-2013) (New usage is discouraged.)

		Ref	Expression
	Assertion	flddivrng	⊢ ( 𝐾 ∈ Fld → 𝐾 ∈ DivRingOps )

Proof

Step	Hyp	Ref	Expression
1		df-fld	⊢ Fld = ( DivRingOps ∩ Com2 )
2		inss1	⊢ ( DivRingOps ∩ Com2 ) ⊆ DivRingOps
3	1 2	eqsstri	⊢ Fld ⊆ DivRingOps
4	3	sseli	⊢ ( 𝐾 ∈ Fld → 𝐾 ∈ DivRingOps )