Theorem mndomgmid 34002

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 34001	. 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ Magma)
2		mndoisexid 34000	. 2 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
3	1, 2	elind 3949	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))

Syntax hints:   → wi 4   ∈ wcel 2145   ∩ cin 3722   ExId cexid 33975  Magmacmagm 33979  MndOpcmndo 33997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-sgrOLD 33992  df-mndo 33998

This theorem is referenced by:  ismndo2  34005  rngoidmlem  34067  isdrngo2  34089