![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > vnex | Structured version Visualization version GIF version |
Description: The universal class does not exist as a set. (Contributed by NM, 4-Jul-2005.) |
Ref | Expression |
---|---|
vnex | ⊢ ¬ ∃𝑥 𝑥 = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nalset 5268 | . 2 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | vex 3447 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | tbt 369 | . . . . 5 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
4 | 3 | albii 1821 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
5 | dfcleq 2730 | . . . 4 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) | |
6 | 4, 5 | bitr4i 277 | . . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V) |
7 | 6 | exbii 1850 | . 2 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V) |
8 | 1, 7 | mtbi 321 | 1 ⊢ ¬ ∃𝑥 𝑥 = V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∀wal 1539 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3445 |
This theorem is referenced by: vprc 5270 |
Copyright terms: Public domain | W3C validator |