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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 5191 | . 2 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | vex 3402 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | tbt 373 | . . . . 5 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
4 | 3 | albii 1827 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
5 | dfcleq 2729 | . . . 4 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) | |
6 | 4, 5 | bitr4i 281 | . . 3 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V) |
7 | 6 | exbii 1855 | . 2 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V) |
8 | 1, 7 | mtbi 325 | 1 ⊢ ¬ ∃𝑥 𝑥 = V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∀wal 1541 = wceq 1543 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 ax-sep 5177 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 |
This theorem is referenced by: vprc 5193 |
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