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Theorem setindel 4201
Description: -Induction in terms of membership in a class. (Contributed by Mario Carneiro and Jim Kingdon, 22-Oct-2018.)
Assertion
Ref Expression
setindel (x(y(y xy 𝑆) → x 𝑆) → 𝑆 = V)
Distinct variable group:   x,y,𝑆

Proof of Theorem setindel
StepHypRef Expression
1 clelsb3 2120 . . . . . . 7 ([y / x]x 𝑆y 𝑆)
21ralbii 2304 . . . . . 6 (y x [y / x]x 𝑆y x y 𝑆)
3 df-ral 2285 . . . . . 6 (y x y 𝑆y(y xy 𝑆))
42, 3bitri 173 . . . . 5 (y x [y / x]x 𝑆y(y xy 𝑆))
54imbi1i 227 . . . 4 ((y x [y / x]x 𝑆x 𝑆) ↔ (y(y xy 𝑆) → x 𝑆))
65albii 1335 . . 3 (x(y x [y / x]x 𝑆x 𝑆) ↔ x(y(y xy 𝑆) → x 𝑆))
7 ax-setind 4200 . . 3 (x(y x [y / x]x 𝑆x 𝑆) → x x 𝑆)
86, 7sylbir 125 . 2 (x(y(y xy 𝑆) → x 𝑆) → x x 𝑆)
9 eqv 3213 . 2 (𝑆 = V ↔ x x 𝑆)
108, 9sylibr 137 1 (x(y(y xy 𝑆) → x 𝑆) → 𝑆 = V)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1224   = wceq 1226   wcel 1370  [wsb 1623  wral 2280  Vcvv 2531
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-ral 2285  df-v 2533
This theorem is referenced by:  setind  4202
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