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Theorem mss 4181
 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 2712 . . . . 5
21snss 3681 . . . 4
31snm 3675 . . . . 5
41snex 4141 . . . . . 6
5 sseq1 3147 . . . . . . 7
6 eleq2 2218 . . . . . . . 8
76exbidv 1802 . . . . . . 7
85, 7anbi12d 465 . . . . . 6
94, 8spcev 2804 . . . . 5
103, 9mpan2 422 . . . 4
112, 10sylbi 120 . . 3
1211exlimiv 1575 . 2
13 elequ1 2129 . . . . 5
1413cbvexv 1895 . . . 4
1514anbi2i 453 . . 3
1615exbii 1582 . 2
1712, 16sylibr 133 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1332  wex 1469   wcel 2125   wss 3098  csn 3556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562 This theorem is referenced by: (None)
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