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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 (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 2554 . . . . 5 y V
21snss 3485 . . . 4 (y A ↔ {y} ⊆ A)
31snm 3479 . . . . 5 w w {y}
4 snexgOLD 3926 . . . . . . 7 (y V → {y} V)
51, 4ax-mp 7 . . . . . 6 {y} V
6 sseq1 2960 . . . . . . 7 (x = {y} → (xA ↔ {y} ⊆ A))
7 eleq2 2098 . . . . . . . 8 (x = {y} → (w xw {y}))
87exbidv 1703 . . . . . . 7 (x = {y} → (w w xw w {y}))
96, 8anbi12d 442 . . . . . 6 (x = {y} → ((xA w w x) ↔ ({y} ⊆ A w w {y})))
105, 9spcev 2641 . . . . 5 (({y} ⊆ A w w {y}) → x(xA w w x))
113, 10mpan2 401 . . . 4 ({y} ⊆ Ax(xA w w x))
122, 11sylbi 114 . . 3 (y Ax(xA w w x))
1312exlimiv 1486 . 2 (y y Ax(xA w w x))
14 elequ1 1597 . . . . 5 (z = w → (z xw x))
1514cbvexv 1792 . . . 4 (z z xw w x)
1615anbi2i 430 . . 3 ((xA z z x) ↔ (xA w w x))
1716exbii 1493 . 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 1242  wex 1378   wcel 1390  Vcvv 2551  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-bnd 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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