Theorem smgrpismgmOLD 38494

Description: Obsolete version of sgrpmgm 18783 as of 3-Feb-2020. A semigroup is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
smgrpismgmOLD	⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Proof of Theorem smgrpismgmOLD

Step	Hyp	Ref	Expression
1		elin 3922	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
2	1	simplbi 501	. 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3		df-sgrOLD 38493	. 2 ⊢ SemiGrp = (Magma ∩ Ass)
4	2, 3	eleq2s 2881	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

Syntax hints:   → wi 4   ∈ wcel 2143   ∩ cin 3905  Asscass 38474  Magmacmagm 38480  SemiGrpcsem 38492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-in 3913  df-sgrOLD 38493

This theorem is referenced by:  mndoismgmOLD  38502