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Theorem ustne0 20444
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 20437 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  U
)
2 ne0i 3784 . 2  |-  ( ( X  X.  X )  e.  U  ->  U  =/=  (/) )
31, 2syl 16 1  |-  ( U  e.  (UnifOn `  X
)  ->  U  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762    =/= wne 2655   (/)c0 3778    X. cxp 4990   ` cfv 5579  UnifOncust 20430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-res 5004  df-iota 5542  df-fun 5581  df-fv 5587  df-ust 20431
This theorem is referenced by:  utopbas  20466  cstucnd  20515
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