Theorem isnmnd 18817

Description: A condition for a structure not to be a monoid: every element of the base set is not a left identity for at least one element of the base set. (Contributed by AV, 4-Feb-2020.)

Hypotheses

Ref	Expression
isnmnd.b	⊢ 𝐵 = (Base‘𝑀)
isnmnd.o	⊢ ⚬ = (+g‘𝑀)

Assertion

Ref	Expression
isnmnd	⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd)

Distinct variable groups:   𝑥,𝐵,𝑧   𝑥,𝑀,𝑧   𝑥, ⚬ ,𝑧

Proof of Theorem isnmnd

Step	Hyp	Ref	Expression
1		neneq 2966	. . . . . . . 8 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ (𝑧 ⚬ 𝑥) = 𝑥)
2	1	intnanrd 495	. . . . . . 7 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
3	2	reximi 3105	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
4	3	ralimi 3104	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
5		rexnal 3119	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
6	5	ralbii 3113	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ∀𝑧 ∈ 𝐵 ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
7		ralnex 3093	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
8	6, 7	bitri 278	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
9	4, 8	sylib 221	. . . 4 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
10	9	intnand 494	. . 3 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ (𝑀 ∈ Smgrp ∧ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥)))
11		isnmnd.b	. . . 4 ⊢ 𝐵 = (Base‘𝑀)
12		isnmnd.o	. . . 4 ⊢ ⚬ = (+g‘𝑀)
13	11, 12	ismnddef 18815	. . 3 ⊢ (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥)))
14	10, 13	sylnibr 332	. 2 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
15		df-nel 3067	. 2 ⊢ (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
16	14, 15	sylibr 237	1 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ≠ wne 2960   ∉ wnel 3066  ∀wral 3081  ∃wrex 3091  ‘cfv 6540  (class class class)co 7416  Basecbs 17286  +_gcplusg 17327  Smgrpcsgrp 18797  Mndcmnd 18813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7419  df-mnd 18814

This theorem is used by:  sgrp2nmndlem5  19014  copisnmnd  48967  nnsgrpnmnd  48976  2zrngnring  49056