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Theorem mss 3953
Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
mss 
C_
Distinct variable groups:   ,   ,   ,,
Allowed substitution hint:   ()

Proof of Theorem mss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 
_V
21snss 3485 . . . 4  { }  C_
31snm 3479 . . . . 5  { }
4 snexgOLD 3926 . . . . . . 7  _V  { }  _V
51, 4ax-mp 7 . . . . . 6  { }  _V
6 sseq1 2960 . . . . . . 7  { }  C_  { }  C_
7 eleq2 2098 . . . . . . . 8  { }  { }
87exbidv 1703 . . . . . . 7  { }  { }
96, 8anbi12d 442 . . . . . 6  { }  C_  { }  C_  { }
105, 9spcev 2641 . . . . 5  { }  C_  { }  C_
113, 10mpan2 401 . . . 4  { }  C_  C_
122, 11sylbi 114 . . 3  C_
1312exlimiv 1486 . 2 
C_
14 elequ1 1597 . . . . 5
1514cbvexv 1792 . . . 4
1615anbi2i 430 . . 3  C_  C_
1716exbii 1493 . 2  C_  C_
1813, 17sylibr 137 1 
C_
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   _Vcvv 2551    C_ wss 2911   {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373
This theorem is referenced by: (None)
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