Theorem isnmgm 18705

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	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
isnmgm	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)

Proof of Theorem isnmgm

Step	Hyp	Ref	Expression
1		mgmcl.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
2		mgmcl.o	. . . . . 6 ⊢ ⚬ = (+g‘𝑀)
3	1, 2	mgmcl 18704	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
4	3	3expib 1138	. . . 4 ⊢ (𝑀 ∈ Mgm → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵))
5	4	com12 33	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀 ∈ Mgm → (𝑋 ⚬ 𝑌) ∈ 𝐵))
6	5	nelcon3d 3075	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ⚬ 𝑌) ∉ 𝐵 → 𝑀 ∉ Mgm))
7	6	3impia 1133	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150   ∉ wnel 3071  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  +_gcplusg 17313  Mgmcmgm 18699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-nul 5274

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ne 2966  df-nel 3072  df-ral 3087  df-rab 3424  df-v 3464  df-sbc 3753  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6496  df-fv 6548  df-ov 7417  df-mgm 18701

This theorem is referenced by:  oddinmgm  48889