Theorem mndomgmid 38119

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 38118	. 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma)
2		mndoisexid 38117	. 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
3	1, 2	elind 4154	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3902   ExId cexid 38092  Magmacmagm 38096  MndOpcmndo 38114

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 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-in 3910  df-sgrOLD 38109  df-mndo 38115

This theorem is referenced by:  ismndo2  38122  rngoidmlem  38184  isdrngo2  38206