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Theorem elnev 39165
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 3359 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 df-v 3353 . . . . 5 V = {𝑥𝑥 = 𝑥}
32eqeq2i 2783 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
4 equid 2097 . . . . . . 7 𝑥 = 𝑥
54tbt 358 . . . . . 6 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴𝑥 = 𝑥))
65albii 1895 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥𝑥 = 𝐴𝑥 = 𝑥))
7 alnex 1854 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
8 abbi 2886 . . . . 5 (∀𝑥𝑥 = 𝐴𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
96, 7, 83bitr3ri 291 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
103, 9bitri 264 . . 3 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
1110necon2abii 2993 . 2 (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
121, 11bitri 264 1 (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1629   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  Vcvv 3351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ne 2944  df-v 3353
This theorem is referenced by: (None)
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