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Theorem setindel 4220
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 2139 . . . . . . 7 ([y / x]x 𝑆y 𝑆)
21ralbii 2324 . . . . . 6 (y x [y / x]x 𝑆y x y 𝑆)
3 df-ral 2305 . . . . . 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 1356 . . 3 (x(y x [y / x]x 𝑆x 𝑆) ↔ x(y(y xy 𝑆) → x 𝑆))
7 ax-setind 4219 . . 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 3234 . 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 1240   = wceq 1242   wcel 1390  [wsb 1642  wral 2300  Vcvv 2551
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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4219
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-ral 2305  df-v 2553
This theorem is referenced by:  setind  4221
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