Description: Obsolete proof of vnex as of 25-Apr-2026. (Contributed by NM, 4-Jul-2005) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | vnexOLD | ⊢ ¬ ∃ 𝑥 𝑥 = V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nalset | ⊢ ¬ ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥 | |
| 2 | vex | ⊢ 𝑦 ∈ V | |
| 3 | 2 | tbt | ⊢ ( 𝑦 ∈ 𝑥 ↔ ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V ) ) |
| 4 | 3 | albii | ⊢ ( ∀ 𝑦 𝑦 ∈ 𝑥 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V ) ) |
| 5 | dfcleq | ⊢ ( 𝑥 = V ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V ) ) | |
| 6 | 4 5 | bitr4i | ⊢ ( ∀ 𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V ) |
| 7 | 6 | exbii | ⊢ ( ∃ 𝑥 ∀ 𝑦 𝑦 ∈ 𝑥 ↔ ∃ 𝑥 𝑥 = V ) |
| 8 | 1 7 | mtbi | ⊢ ¬ ∃ 𝑥 𝑥 = V |