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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 5311 | . . 3 ⊢ ∅ ∈ V | |
2 | 0nelxp 5716 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
3 | nelss 4047 | . . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 690 | . 2 ⊢ ¬ V ⊆ (V × V) |
5 | df-rel 5689 | . 2 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
6 | 4, 5 | mtbir 322 | 1 ⊢ ¬ Rel V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 ∅c0 4326 × cxp 5680 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-opab 5215 df-xp 5688 df-rel 5689 |
This theorem is referenced by: nfunv 6591 relintabex 43042 |
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