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Theorem isnmnd 17121
 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 = (+g𝑀)
Assertion
Ref Expression
isnmnd (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)
Distinct variable groups:   𝑥,𝐵,𝑧   𝑥,𝑀,𝑧   𝑥, ,𝑧

Proof of Theorem isnmnd
StepHypRef Expression
1 df-ne 2782 . . . . . . . . 9 ((𝑧 𝑥) ≠ 𝑥 ↔ ¬ (𝑧 𝑥) = 𝑥)
21biimpi 205 . . . . . . . 8 ((𝑧 𝑥) ≠ 𝑥 → ¬ (𝑧 𝑥) = 𝑥)
32intnanrd 954 . . . . . . 7 ((𝑧 𝑥) ≠ 𝑥 → ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
43reximi 2994 . . . . . 6 (∃𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
54ralimi 2936 . . . . 5 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
6 rexnal 2978 . . . . . . 7 (∃𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
76ralbii 2963 . . . . . 6 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
8 ralnex 2975 . . . . . 6 (∀𝑧𝐵 ¬ ∀𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
97, 8bitri 263 . . . . 5 (∀𝑧𝐵𝑥𝐵 ¬ ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥) ↔ ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
105, 9sylib 207 . . . 4 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥))
1110intnand 953 . . 3 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ (𝑀 ∈ SGrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
12 isnmnd.b . . . 4 𝐵 = (Base‘𝑀)
13 isnmnd.o . . . 4 = (+g𝑀)
1412, 13ismnddef 17119 . . 3 (𝑀 ∈ Mnd ↔ (𝑀 ∈ SGrp ∧ ∃𝑧𝐵𝑥𝐵 ((𝑧 𝑥) = 𝑥 ∧ (𝑥 𝑧) = 𝑥)))
1511, 14sylnibr 318 . 2 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥 → ¬ 𝑀 ∈ Mnd)
16 df-nel 2783 . 2 (𝑀 ∉ Mnd ↔ ¬ 𝑀 ∈ Mnd)
1715, 16sylibr 223 1 (∀𝑧𝐵𝑥𝐵 (𝑧 𝑥) ≠ 𝑥𝑀 ∉ Mnd)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  SGrpcsgrp 17106  Mndcmnd 17117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-mnd 17118 This theorem is referenced by:  sgrp2nmndlem5  17239  copisnmnd  41599  nnsgrpnmnd  41608  2zrngnring  41742
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