Theorem smgrpismgmOLD 38999

Description: Obsolete version of sgrpmgm 18945 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 4096	. . 3 ⊢ (𝐺 ∈ (Magma ∩ Ass) ↔ (𝐺 ∈ Magma ∧ 𝐺 ∈ Ass))
2	1	simplbi 502	. 2 ⊢ (𝐺 ∈ (Magma ∩ Ass) → 𝐺 ∈ Magma)
3		df-sgrOLD 38998	. 2 ⊢ SemiGrp = (Magma ∩ Ass)
4	2, 3	eleq2s 2882	1 ⊢ (𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092  Asscass 38979  Magmacmagm 38985  SemiGrpcsem 38997

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-sgrOLD 38998

This theorem is used by:  mndoismgmOLD  39007