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Mirrors > Home > MPE Home > Th. List > nrelv | Structured version Visualization version GIF version |
Description: The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.) |
Ref | Expression |
---|---|
nrelv | ⊢ ¬ Rel V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5313 | . . 3 ⊢ ∅ ∈ V | |
2 | 0nelxp 5723 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
3 | nelss 4061 | . . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ ¬ V ⊆ (V × V) |
5 | df-rel 5696 | . 2 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
6 | 4, 5 | mtbir 323 | 1 ⊢ ¬ Rel V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 × cxp 5687 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: nfunv 6601 relintabex 43571 |
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