Theorem setindel 4574

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 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝑆 = V)

Distinct variable group:   𝑥,𝑦,𝑆

Proof of Theorem setindel

Step	Hyp	Ref	Expression
1		clelsb1 2301	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆)
2	1	ralbii 2503	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑆)
3		df-ral 2480	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑆 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆))
4	2, 3	bitri 184	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆))
5	4	imbi1i 238	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) ↔ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆))
6	5	albii 1484	. . 3 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆))
7		ax-setind 4573	. . 3 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) → ∀𝑥 𝑥 ∈ 𝑆)
8	6, 7	sylbir 135	. 2 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → ∀𝑥 𝑥 ∈ 𝑆)
9		eqv 3470	. 2 ⊢ (𝑆 = V ↔ ∀𝑥 𝑥 ∈ 𝑆)
10	8, 9	sylibr 134	1 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝑆 = V)

Syntax hints:   → wi 4  ∀wal 1362   = wceq 1364  [wsb 1776   ∈ wcel 2167  ∀wral 2475  Vcvv 2763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-v 2765

This theorem is referenced by:  setind  4575