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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 5272 | . 2 ⊢ ¬ ∃𝑥 𝑥 = V | |
2 | isset 3459 | . 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 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3446 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3448 |
This theorem is referenced by: nvel 5274 intex 5295 intnex 5296 abnex 7692 iprc 7851 opabn1stprc 7991 elfi2 9351 fi0 9357 ruALT 9540 cardmin2 9936 00lsp 20445 n0lplig 29428 fveqvfvv 45281 ndmaovcl 45442 vsn 46903 |
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