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Theorem isnmgm 17632
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 17631 . . . . 5 ((𝑀 ∈ Mgm ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵)
433expib 1113 . . . 4 (𝑀 ∈ Mgm → ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) ∈ 𝐵))
54com12 32 . . 3 ((𝑋𝐵𝑌𝐵) → (𝑀 ∈ Mgm → (𝑋 𝑌) ∈ 𝐵))
65nelcon3d 3087 . 2 ((𝑋𝐵𝑌𝐵) → ((𝑋 𝑌) ∉ 𝐵𝑀 ∉ Mgm))
763impia 1106 1 ((𝑋𝐵𝑌𝐵 ∧ (𝑋 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wnel 3075  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  Mgmcmgm 17626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-nel 3076  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-mgm 17628
This theorem is referenced by:  oddinmgm  42830
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