Theorem isnmnd 18672

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 2932	. . . . . . . 8 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ (𝑧 ⚬ 𝑥) = 𝑥)
2	1	intnanrd 489	. . . . . . 7 ⊢ ((𝑧 ⚬ 𝑥) ≠ 𝑥 → ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
3	2	reximi 3068	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
4	3	ralimi 3067	. . . . 5 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑧 ⚬ 𝑥) ≠ 𝑥 → ∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
5		rexnal 3083	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
6	5	ralbii 3076	. . . . . 6 ⊢ (∀𝑧 ∈ 𝐵 ∃𝑥 ∈ 𝐵 ¬ ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥) ↔ ∀𝑧 ∈ 𝐵 ¬ ∀𝑥 ∈ 𝐵 ((𝑧 ⚬ 𝑥) = 𝑥 ∧ (𝑥 ⚬ 𝑧) = 𝑥))
7		ralnex 3056	. . . . . 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 18670	. . 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 3031	. 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 1540   ∈ wcel 2109   ≠ wne 2926   ∉ wnel 3030  ∀wral 3045  ∃wrex 3054  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Smgrpcsgrp 18652  Mndcmnd 18668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-nul 5264

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ov 7393  df-mnd 18669

This theorem is referenced by:  sgrp2nmndlem5  18863  copisnmnd  48161  nnsgrpnmnd  48170  2zrngnring  48250