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Theorem wunelss 9151
Description: The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1  |-  ( ph  ->  U  e. WUni )
wununi.2  |-  ( ph  ->  A  e.  U )
Assertion
Ref Expression
wunelss  |-  ( ph  ->  A  C_  U )

Proof of Theorem wunelss
StepHypRef Expression
1 wununi.1 . . 3  |-  ( ph  ->  U  e. WUni )
2 wuntr 9148 . . 3  |-  ( U  e. WUni  ->  Tr  U )
31, 2syl 17 . 2  |-  ( ph  ->  Tr  U )
4 wununi.2 . 2  |-  ( ph  ->  A  e.  U )
5 trss 4499 . 2  |-  ( Tr  U  ->  ( A  e.  U  ->  A  C_  U ) )
63, 4, 5sylc 61 1  |-  ( ph  ->  A  C_  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1904    C_ wss 3390   Tr wtr 4490  WUnicwun 9143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-in 3397  df-ss 3404  df-uni 4191  df-tr 4491  df-wun 9145
This theorem is referenced by:  wunss  9155  wunf  9170  wuncval2  9190
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