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Theorem isnmgm 17856
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 17855 . . . . 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 3130 . 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 399  w3a 1084   = wceq 1538  wcel 2115  wnel 3118  cfv 6343  (class class class)co 7149  Basecbs 16483  +gcplusg 16565  Mgmcmgm 17850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-nel 3119  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7152  df-mgm 17852
This theorem is referenced by:  oddinmgm  44361
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