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Theorem elnev 41729
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 3421 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2 df-v 3410 . . . . 5 V = {𝑥𝑥 = 𝑥}
32eqeq2i 2750 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
4 equid 2020 . . . . . . 7 𝑥 = 𝑥
54tbt 373 . . . . . 6 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴𝑥 = 𝑥))
65albii 1827 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥𝑥 = 𝐴𝑥 = 𝑥))
7 alnex 1789 . . . . 5 (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
8 abbi 2810 . . . . 5 (∀𝑥𝑥 = 𝐴𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥})
96, 7, 83bitr3ri 305 . . . 4 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
103, 9bitri 278 . . 3 ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
1110necon2abii 2991 . 2 (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
121, 11bitri 278 1 (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wal 1541   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wne 2940  Vcvv 3408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3410
This theorem is referenced by: (None)
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