Theorem vnexOLD 5275

Description: Obsolete proof of vnex 5274 as of 25-Apr-2026. (Contributed by NM, 4-Jul-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
vnexOLD	⊢ ¬ ∃𝑥 𝑥 = V

Proof of Theorem vnexOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nalset 5271	. 2 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	tbt 372	. . . . 5 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1852	. . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
5		dfcleq 2756	. . . 4 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
6	4, 5	bitr4i 281	. . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V)
7	6	exbii 1881	. 2 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 325	1 ⊢ ¬ ∃𝑥 𝑥 = V

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by: (None)