Theorem isnmgm 18632

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 18631	. . . . 5 ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
4	3	3expib 1119	. . . 4 ⊢ (𝑀 ∈ Mgm → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵))
5	4	com12 32	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀 ∈ Mgm → (𝑋 ⚬ 𝑌) ∈ 𝐵))
6	5	nelcon3d 3040	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ⚬ 𝑌) ∉ 𝐵 → 𝑀 ∉ Mgm))
7	6	3impia 1114	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ∉ wnel 3036  ‘cfv 6546  (class class class)co 7416  Basecbs 17208  +_gcplusg 17261  Mgmcmgm 18626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5303

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-nel 3037  df-ral 3052  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-iota 6498  df-fv 6554  df-ov 7419  df-mgm 18628

This theorem is referenced by:  oddinmgm  47588