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Theorem mss 3936
 Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
mss (y y Ax(xA z z x))
Distinct variable groups:   x,y   x,z   x,A,y
Allowed substitution hint:   A(z)

Proof of Theorem mss
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 vex 2538 . . . . 5 y V
21snss 3468 . . . 4 (y A ↔ {y} ⊆ A)
31snm 3462 . . . . 5 w w {y}
4 snexgOLD 3909 . . . . . . 7 (y V → {y} V)
51, 4ax-mp 7 . . . . . 6 {y} V
6 sseq1 2943 . . . . . . 7 (x = {y} → (xA ↔ {y} ⊆ A))
7 eleq2 2083 . . . . . . . 8 (x = {y} → (w xw {y}))
87exbidv 1688 . . . . . . 7 (x = {y} → (w w xw w {y}))
96, 8anbi12d 445 . . . . . 6 (x = {y} → ((xA w w x) ↔ ({y} ⊆ A w w {y})))
105, 9spcev 2624 . . . . 5 (({y} ⊆ A w w {y}) → x(xA w w x))
113, 10mpan2 403 . . . 4 ({y} ⊆ Ax(xA w w x))
122, 11sylbi 114 . . 3 (y Ax(xA w w x))
1312exlimiv 1471 . 2 (y y Ax(xA w w x))
14 elequ1 1582 . . . . 5 (z = w → (z xw x))
1514cbvexv 1777 . . . 4 (z z xw w x)
1615anbi2i 433 . . 3 ((xA z z x) ↔ (xA w w x))
1716exbii 1478 . 2 (x(xA z z x) ↔ x(xA w w x))
1813, 17sylibr 137 1 (y y Ax(xA z z x))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374  Vcvv 2535   ⊆ wss 2894  {csn 3350 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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356 This theorem is referenced by: (None)
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