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Mirrors > Home > MPE Home > Th. List > vprc | Structured version Visualization version GIF version |
Description: The universal class is not a member of itself (and thus is not a set). Proposition 5.21 of [TakeutiZaring] p. 21; our proof, however, does not depend on the Axiom of Regularity. (Contributed by NM, 23-Aug-1993.) |
Ref | Expression |
---|---|
vprc | ⊢ ¬ V ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nalset 4723 | . . 3 ⊢ ¬ ∃𝑥∀𝑦 𝑦 ∈ 𝑥 | |
2 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 2 | tbt 358 | . . . . . 6 ⊢ (𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
4 | 3 | albii 1737 | . . . . 5 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) |
5 | dfcleq 2604 | . . . . 5 ⊢ (𝑥 = V ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ∈ V)) | |
6 | 4, 5 | bitr4i 266 | . . . 4 ⊢ (∀𝑦 𝑦 ∈ 𝑥 ↔ 𝑥 = V) |
7 | 6 | exbii 1764 | . . 3 ⊢ (∃𝑥∀𝑦 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝑥 = V) |
8 | 1, 7 | mtbi 311 | . 2 ⊢ ¬ ∃𝑥 𝑥 = V |
9 | isset 3180 | . 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V) | |
10 | 8, 9 | mtbir 312 | 1 ⊢ ¬ V ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∀wal 1473 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 |
This theorem depends on definitions: df-bi 196 df-an 385 df-tru 1478 df-ex 1696 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: nvel 4725 vnex 4726 intex 4747 intnex 4748 snnex 6862 iprc 6993 elfi2 8203 fi0 8209 ruALT 8391 cardmin2 8707 00lsp 18802 bj-xnex 32245 fveqvfvv 39853 ndmaovcl 39932 opabn1stprc 40318 |
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