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Theorem 0mgm 46154
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 (𝑀𝑉𝑀 ∈ Mgm)

Proof of Theorem 0mgm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 4471 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅
2 0mgm.b . . . 4 (Base‘𝑀) = ∅
32eqcomi 2742 . . 3 ∅ = (Base‘𝑀)
4 eqid 2733 . . 3 (+g𝑀) = (+g𝑀)
53, 4ismgm 18503 . 2 (𝑀𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅))
61, 5mpbiri 258 1 (𝑀𝑉𝑀 ∈ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3061  c0 4283  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Mgmcmgm 18500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-ov 7361  df-mgm 18502
This theorem is referenced by: (None)
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