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Theorem elnev 40650
Description: Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
Assertion
Ref Expression
elnev (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
Distinct variable group:   𝑥,𝐴

Proof of Theorem elnev
StepHypRef Expression
1 isset 3507 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 df-v 3497 . . . . 5 V = {𝑥𝑥 = 𝑥}
32eqeq2i 2834 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
4 equid 2010 . . . . . . 7 𝑥 = 𝑥
54tbt 371 . . . . . 6 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴𝑥 = 𝑥))
65albii 1811 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥𝑥 = 𝐴𝑥 = 𝑥))
7 alnex 1773 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
8 abbi 2888 . . . . 5 (∀𝑥𝑥 = 𝐴𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
96, 7, 83bitr3ri 303 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
103, 9bitri 276 . . 3 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
1110necon2abii 3066 . 2 (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
121, 11bitri 276 1 (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wal 1526   = wceq 1528  wex 1771  wcel 2105  {cab 2799  wne 3016  Vcvv 3495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-v 3497
This theorem is referenced by: (None)
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