Theorem elnev 44961

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 3462	. 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		df-v 3450	. . . . 5 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥}
3	2	eqeq2i 2769	. . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥})
4		abbib 2825	. . . . 5 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
5		equid 2026	. . . . . . 7 ⊢ 𝑥 = 𝑥
6	5	tbt 371	. . . . . 6 ⊢ (¬ 𝑥 = 𝐴 ↔ (¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
7	6	albii 1833	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ∀𝑥(¬ 𝑥 = 𝐴 ↔ 𝑥 = 𝑥))
8		alnex 1795	. . . . 5 ⊢ (∀𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃𝑥 𝑥 = 𝐴)
9	4, 7, 8	3bitr2i 301	. . . 4 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = {𝑥 ∣ 𝑥 = 𝑥} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
10	3, 9	bitri 277	. . 3 ⊢ ({𝑥 ∣ ¬ 𝑥 = 𝐴} = V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
11	10	necon2abii 3001	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)
12	1, 11	bitri 277	1 ⊢ (𝐴 ∈ V ↔ {𝑥 ∣ ¬ 𝑥 = 𝐴} ≠ V)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1552   = wceq 1554  ∃wex 1793   ∈ wcel 2136  {cab 2734   ≠ wne 2951  Vcvv 3448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ne 2952  df-v 3450

This theorem is referenced by: (None)