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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 | vnex 5314 | . 2 ⊢ ¬ ∃𝑥 𝑥 = V | |
| 2 | isset 3494 | . 2 ⊢ (V ∈ V ↔ ∃𝑥 𝑥 = V) | |
| 3 | 1, 2 | mtbir 323 | 1 ⊢ ¬ V ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 |
| This theorem is referenced by: nvel 5316 intex 5344 intnex 5345 abnex 7777 iprc 7933 opabn1stprc 8083 elfi2 9454 fi0 9460 ruALT 9643 cardmin2 10039 00lsp 20979 n0lplig 30502 fveqvfvv 47052 ndmaovcl 47215 vsn 48731 |
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