Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5231 | . . 3 ⊢ ∅ ∈ V | |
2 | 0nelxp 5623 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
3 | nelss 3984 | . . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 689 | . 2 ⊢ ¬ V ⊆ (V × V) |
5 | df-rel 5596 | . 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 3432 ⊆ wss 3887 ∅c0 4256 × cxp 5587 Rel wrel 5594 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5137 df-xp 5595 df-rel 5596 |
This theorem is referenced by: nfunv 6467 relintabex 41189 |
Copyright terms: Public domain | W3C validator |