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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
StepHypRef Expression
1 mndoismgmOLD 34001 . 2 (𝐺 ∈ MndOp → 𝐺 ∈ Magma)
2 mndoisexid 34000 . 2 (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
31, 2elind 3949 1 (𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
 Colors of variables: wff setvar class 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
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