Theorem mndomgmid 38072

Description: A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)

Assertion

Ref	Expression
mndomgmid	⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))

Proof of Theorem mndomgmid

Step	Hyp	Ref	Expression
1		mndoismgmOLD 38071	. 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma)
2		mndoisexid 38070	. 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
3	1, 2	elind 4152	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))

Syntax hints:   → wi 4   ∈ wcel 2113   ∩ cin 3900   ExId cexid 38045  Magmacmagm 38049  MndOpcmndo 38067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-in 3908  df-sgrOLD 38062  df-mndo 38068

This theorem is referenced by:  ismndo2  38075  rngoidmlem  38137  isdrngo2  38159