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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 5026 | . . 3 ⊢ ∅ ∈ V | |
2 | 0nelxp 5389 | . . 3 ⊢ ¬ ∅ ∈ (V × V) | |
3 | nelss 3882 | . . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V)) | |
4 | 1, 2, 3 | mp2an 682 | . 2 ⊢ ¬ V ⊆ (V × V) |
5 | df-rel 5362 | . 2 ⊢ (Rel V ↔ V ⊆ (V × V)) | |
6 | 4, 5 | mtbir 315 | 1 ⊢ ¬ Rel V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2106 Vcvv 3397 ⊆ wss 3791 ∅c0 4140 × cxp 5353 Rel wrel 5360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-opab 4949 df-xp 5361 df-rel 5362 |
This theorem is referenced by: nfunv 6168 |
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