Theorem vnexOLD 5282

Description: Obsolete proof of vnex 5281 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 5278	. 2 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥
2		vex 3459	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	tbt 372	. . . . 5 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V))
4	3	albii 1849	. . . 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 1878	. 2 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V)
8	1, 7	mtbi 325	1 ⊢ ¬ ∃𝑥 𝑥 = V

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457

This theorem is referenced by: (None)