Theorem 0mgm 48793

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 4454	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅
2		0mgm.b	. . . 4 ⊢ (Base‘𝑀) = ∅
3	2	eqcomi 2773	. . 3 ⊢ ∅ = (Base‘𝑀)
4		eqid 2764	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
5	3, 4	ismgm 18677	. 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 1562   ∈ wcel 2144  ∀wral 3078  ∅c0 4287  ‘cfv 6523  (class class class)co 7398  Basecbs 17247  +_gcplusg 17288  Mgmcmgm 18674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531  df-ov 7401  df-mgm 18676

This theorem is referenced by: (None)