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Theorem 0mgm 47219
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 4509 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g𝑀)𝑦) ∈ ∅
2 0mgm.b . . . 4 (Base‘𝑀) = ∅
32eqcomi 2737 . . 3 ∅ = (Base‘𝑀)
4 eqid 2728 . . 3 (+g𝑀) = (+g𝑀)
53, 4ismgm 18595 . 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 1534  wcel 2099  wral 3057  c0 4319  cfv 6543  (class class class)co 7415  Basecbs 17174  +gcplusg 17227  Mgmcmgm 18592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5301
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-iota 6495  df-fv 6551  df-ov 7418  df-mgm 18594
This theorem is referenced by: (None)
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