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Theorem ustne0 21006
Description: A uniform structure cannot be empty. (Contributed by Thierry Arnoux, 16-Nov-2017.)
Assertion
Ref Expression
ustne0  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )

Proof of Theorem ustne0
StepHypRef Expression
1 ustbasel 20999 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
2 ne0i 3743 . 2  |-  ( ( X  X.  X )  e.  U  ->  U  =/=  (/) )
31, 2syl 17 1  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1842    =/= wne 2598   (/)c0 3737    X. cxp 4820   ` cfv 5568  UnifOncust 20992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-res 4834  df-iota 5532  df-fun 5570  df-fv 5576  df-ust 20993
This theorem is referenced by:  utopbas  21028  cstucnd  21077
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