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Theorem smgrpismgm 21912
Description: A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
smgrpismgm  |-  ( G  e.  SemiGrp  ->  G  e.  Magma )

Proof of Theorem smgrpismgm
StepHypRef Expression
1 elin 3522 . . 3  |-  ( G  e.  ( Magma  i^i  Ass ) 
<->  ( G  e.  Magma  /\  G  e.  Ass )
)
21simplbi 447 . 2  |-  ( G  e.  ( Magma  i^i  Ass )  ->  G  e.  Magma )
3 df-sgr 21911 . 2  |-  SemiGrp  =  (
Magma  i^i  Ass )
42, 3eleq2s 2527 1  |-  ( G  e.  SemiGrp  ->  G  e.  Magma )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    i^i cin 3311   Asscass 21892   Magmacmagm 21898   SemiGrpcsem 21910
This theorem is referenced by:  mndoismgm  21921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-sgr 21911
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