Theorem nev 44610

Description: Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)

Assertion

Ref	Expression
nev	⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem nev

Step	Hyp	Ref	Expression
1		eqv 3460	. 2 ⊢ (𝐴 = V ↔ ∀𝑥 𝑥 ∈ 𝐴)
2	1	necon3abii 3001	1 ⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-v 3452

This theorem is used by: (None)