Theorem 0mgm 49263

Description: A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.)

Hypothesis

Ref	Expression
0mgm.b	⊢ (Base‘𝑀) = ∅

Assertion

Ref	Expression
0mgm	⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)

Proof of Theorem 0mgm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4453	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅
2		0mgm.b	. . . 4 ⊢ (Base‘𝑀) = ∅
3	2	eqcomi 2769	. . 3 ⊢ ∅ = (Base‘𝑀)
4		eqid 2760	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
5	3, 4	ismgm 18817	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅))
6	1, 5	mpbiri 261	1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∅c0 4278  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  Mgmcmgm 18814

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-nul 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-iota 6494  df-fv 6546  df-ov 7423  df-mgm 18816

This theorem is used by: (None)