Theorem elnev 44450

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 3495	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		df-v 3483	. . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
3	2	eqeq2i 2750	. . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥})
4		abbib 2811	. . . . 5 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
5		equid 2011	. . . . . . 7 ⊢ 𝑥 = 𝑥
6	5	tbt 369	. . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
7	6	albii 1818	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
8		alnex 1780	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
9	4, 7, 8	3bitr2i 299	. . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
10	3, 9	bitri 275	. . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
11	10	necon2abii 2991	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
12	1, 11	bitri 275	1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2108  {cab 2714   ≠ wne 2940  Vcvv 3481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3483

This theorem is referenced by: (None)