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Theorem isnmgm 17227
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 17226 . . . . 5 ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
433expib 1266 . . . 4 (𝑀 ∈ Mgm → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
54com12 32 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑀 ∈ Mgm → (𝑋 𝑌) ∈ 𝐵))
65nelcon3d 2906 . 2 ((𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) ∉ 𝐵𝑀 ∉ Mgm))
763impia 1259 1 ((𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  wnel 2894  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  Mgmcmgm 17221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-iota 5839  df-fv 5884  df-ov 6638  df-mgm 17223
This theorem is referenced by:  oddinmgm  41580
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