![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ustne0 | Structured version Visualization version GIF version |
Description: A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.) |
Ref | Expression |
---|---|
ustne0 | ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ustbasel 22498 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈) | |
2 | 1 | ne0d 4223 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2080 ≠ wne 2983 ∅c0 4213 × cxp 5444 ‘cfv 6228 UnifOncust 22491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-res 5458 df-iota 6192 df-fun 6230 df-fv 6236 df-ust 22492 |
This theorem is referenced by: utopbas 22527 cstucnd 22576 |
Copyright terms: Public domain | W3C validator |