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Theorem setindel 4421
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(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → 𝑆 = V)
Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem setindel
StepHypRef Expression
1 clelsb3 2220 . . . . . . 7 ([𝑦 / 𝑥]𝑥𝑆𝑦𝑆)
21ralbii 2416 . . . . . 6 (∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆 ↔ ∀𝑦𝑥 𝑦𝑆)
3 df-ral 2396 . . . . . 6 (∀𝑦𝑥 𝑦𝑆 ↔ ∀𝑦(𝑦𝑥𝑦𝑆))
42, 3bitri 183 . . . . 5 (∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆 ↔ ∀𝑦(𝑦𝑥𝑦𝑆))
54imbi1i 237 . . . 4 ((∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆𝑥𝑆) ↔ (∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆))
65albii 1429 . . 3 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆𝑥𝑆) ↔ ∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆))
7 ax-setind 4420 . . 3 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥]𝑥𝑆𝑥𝑆) → ∀𝑥 𝑥𝑆)
86, 7sylbir 134 . 2 (∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → ∀𝑥 𝑥𝑆)
9 eqv 3350 . 2 (𝑆 = V ↔ ∀𝑥 𝑥𝑆)
108, 9sylibr 133 1 (∀𝑥(∀𝑦(𝑦𝑥𝑦𝑆) → 𝑥𝑆) → 𝑆 = V)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1312   = wceq 1314  wcel 1463  [wsb 1718  wral 2391  Vcvv 2658
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-ral 2396  df-v 2660
This theorem is referenced by:  setind  4422
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