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Theorem isnmgm 18573
Description: A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
Hypotheses
Ref Expression
mgmcl.b 𝐵 = (Base‘𝑀)
mgmcl.o = (+g𝑀)
Assertion
Ref Expression
isnmgm ((𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)

Proof of Theorem isnmgm
StepHypRef Expression
1 mgmcl.b . . . . . 6 𝐵 = (Base‘𝑀)
2 mgmcl.o . . . . . 6 = (+g𝑀)
31, 2mgmcl 18572 . . . . 5 ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
433expib 1119 . . . 4 (𝑀 ∈ Mgm → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
54com12 32 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑀 ∈ Mgm → (𝑋 𝑌) ∈ 𝐵))
65nelcon3d 3042 . 2 ((𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) ∉ 𝐵𝑀 ∉ Mgm))
763impia 1114 1 ((𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wnel 3038  cfv 6534  (class class class)co 7402  Basecbs 17149  +gcplusg 17202  Mgmcmgm 18567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-nel 3039  df-ral 3054  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-ov 7405  df-mgm 18569
This theorem is referenced by:  oddinmgm  47099
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