Theorem elnev 39594

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

Step	Hyp	Ref	Expression
1		isset 3409	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		df-v 3400	. . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
3	2	eqeq2i 2790	. . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥})
4		equid 2059	. . . . . . 7 ⊢ 𝑥 = 𝑥
5	4	tbt 361	. . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
6	5	albii 1863	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
7		alnex 1825	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
8		abbi 2902	. . . . 5 ⊢ (∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥) ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥})
9	6, 7, 8	3bitr3ri 294	. . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
10	3, 9	bitri 267	. . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
11	10	necon2abii 3019	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
12	1, 11	bitri 267	1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)

Syntax hints:  ¬ wn 3   ↔ wb 198  ∀wal 1599   = wceq 1601  ∃wex 1823   ∈ wcel 2107  {cab 2763   ≠ wne 2969  Vcvv 3398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-ne 2970  df-v 3400

This theorem is referenced by: (None)