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Theorem elnev 37558
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 3084 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 df-v 3079 . . . . 5 V = {𝑥𝑥 = 𝑥}
32eqeq2i 2526 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
4 equid 1889 . . . . . . 7 𝑥 = 𝑥
54tbt 357 . . . . . 6 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴𝑥 = 𝑥))
65albii 1722 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥𝑥 = 𝐴𝑥 = 𝑥))
7 alnex 1696 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
8 abbi 2628 . . . . 5 (∀𝑥𝑥 = 𝐴𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
96, 7, 83bitr3ri 289 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
103, 9bitri 262 . . 3 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
1110necon2abii 2736 . 2 (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
121, 11bitri 262 1 (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wal 1472   = wceq 1474  wex 1694  wcel 1938  {cab 2500  wne 2684  Vcvv 3077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494
This theorem depends on definitions:  df-bi 195  df-an 384  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-ne 2686  df-v 3079
This theorem is referenced by: (None)
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