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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 23680 | . 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈) | |
2 | 1 | ne0d 4333 | 1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ≠ wne 2941 ∅c0 4320 × cxp 5670 ‘cfv 6535 UnifOncust 23673 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6487 df-fun 6537 df-fv 6543 df-ust 23674 |
This theorem is referenced by: utopbas 23709 cstucnd 23758 |
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