Theorem 0mgm 48669

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 4428	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅
2		0mgm.b	. . . 4 ⊢ (Base‘𝑀) = ∅
3	2	eqcomi 2750	. . 3 ⊢ ∅ = (Base‘𝑀)
4		eqid 2741	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
5	3, 4	ismgm 18604	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅))
6	1, 5	mpbiri 260	1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∅c0 4263  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  +_gcplusg 17215  Mgmcmgm 18601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5230

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-ral 3056  df-rab 3394  df-v 3435  df-sbc 3725  df-dif 3887  df-un 3889  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-iota 6444  df-fv 6496  df-ov 7362  df-mgm 18603

This theorem is referenced by: (None)