Theorem isnmnd 18721

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 2937	. . . . . . . 8 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ (𝑧 ⚬ 𝑥) = 𝑥)
2	1	intnanrd 489	. . . . . . 7 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
3	2	reximi 3073	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
4	3	ralimi 3072	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
5		rexnal 3088	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
6	5	ralbii 3081	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ∀𝑧 ∈ 𝐵 ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
7		ralnex 3061	. . . . . 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 275	. . . . 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 218	. . . 4 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
10	9	intnand 488	. . 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 18719	. . 3 ⊢ (𝑀 ∈ Mnd ↔ (𝑀 ∈ Smgrp ∧ ∃𝑧 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥)))
14	10, 13	sylnibr 329	. 2 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
15		df-nel 3036	. 2 ⊢ (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
16	14, 15	sylibr 234	1 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → 𝑀 ∉ Mnd)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2931   ∉ wnel 3035  ∀wral 3050  ∃wrex 3059  ‘cfv 6541  (class class class)co 7413  Basecbs 17230  +_gcplusg 17274  Smgrpcsgrp 18701  Mndcmnd 18717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-nul 5286

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-ov 7416  df-mnd 18718

This theorem is referenced by:  sgrp2nmndlem5  18912  copisnmnd  48058  nnsgrpnmnd  48067  2zrngnring  48147