Theorem smgrpismgmOLD 33661

Description: Obsolete version of sgrpmgm 17289 as of 3-Feb-2020. A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
smgrpismgmOLD	|- ( G e. SemiGrp -> G e. Magma )

Proof of Theorem smgrpismgmOLD

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( G e. ( Magma i^i Ass ) <-> ( G e. Magma /\ G e. Ass ) )
2	1	simplbi 476	. 2 |- ( G e. ( Magma i^i Ass ) -> G e. Magma )
3		df-sgrOLD 33660	. 2 |- SemiGrp = ( Magma i^i Ass )
4	2, 3	eleq2s 2719	1 |- ( G e. SemiGrp -> G e. Magma )

Syntax hints:   -> wi 4   e. wcel 1990   i^i cin 3573   Asscass 33641   Magmacmagm 33647   SemiGrpcsem 33659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-sgrOLD 33660

This theorem is referenced by:  mndoismgmOLD  33669