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Theorem mss 3989
 Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
mss
Distinct variable groups:   ,   ,   ,,
Allowed substitution hint:   ()

Proof of Theorem mss
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2605 . . . . 5
21snss 3524 . . . 4
31snm 3518 . . . . 5
41snex 3965 . . . . . 6
5 sseq1 3021 . . . . . . 7
6 eleq2 2143 . . . . . . . 8
76exbidv 1747 . . . . . . 7
85, 7anbi12d 457 . . . . . 6
94, 8spcev 2693 . . . . 5
103, 9mpan2 416 . . . 4
112, 10sylbi 119 . . 3
1211exlimiv 1530 . 2
13 elequ1 1641 . . . . 5
1413cbvexv 1837 . . . 4
1514anbi2i 445 . . 3
1615exbii 1537 . 2
1712, 16sylibr 132 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wceq 1285  wex 1422   wcel 1434   wss 2974  csn 3406 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412 This theorem is referenced by: (None)
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