Theorem 0mgm 48290

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 4462	. 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅
2		0mgm.b	. . . 4 ⊢ (Base‘𝑀) = ∅
3	2	eqcomi 2742	. . 3 ⊢ ∅ = (Base‘𝑀)
4		eqid 2733	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
5	3, 4	ismgm 18551	. 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 1541   ∈ wcel 2113  ∀wral 3048  ∅c0 4282  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  +_gcplusg 17163  Mgmcmgm 18548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-nul 5246

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-iota 6442  df-fv 6494  df-ov 7355  df-mgm 18550

This theorem is referenced by: (None)