Theorem 0mgm 48010

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 4519	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅
2		0mgm.b	. . . 4 ⊢ (Base‘𝑀) = ∅
3	2	eqcomi 2744	. . 3 ⊢ ∅ = (Base‘𝑀)
4		eqid 2735	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
5	3, 4	ismgm 18667	. 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅))
6	1, 5	mpbiri 258	1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∅c0 4339  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  Mgmcmgm 18664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-mgm 18666

This theorem is referenced by: (None)