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Theorem ustne0 22345
Description: A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ustne0 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)

Proof of Theorem ustne0
StepHypRef Expression
1 ustbasel 22338 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
21ne0d 4122 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wne 2971  c0 4115   × cxp 5310  cfv 6101  UnifOncust 22331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-ust 22332
This theorem is referenced by:  utopbas  22367  cstucnd  22416
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