Theorem setindel 4281

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		clelsb3 2183	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆)
2	1	ralbii 2372	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑆)
3		df-ral 2353	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑆 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆))
4	2, 3	bitri 182	. . . . 5 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆))
5	4	imbi1i 236	. . . 4 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) ↔ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆))
6	5	albii 1399	. . 3 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆))
7		ax-setind 4280	. . 3 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) → ∀𝑥 𝑥 ∈ 𝑆)
8	6, 7	sylbir 133	. 2 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → ∀𝑥 𝑥 ∈ 𝑆)
9		eqv 3267	. 2 ⊢ (𝑆 = V ↔ ∀𝑥 𝑥 ∈ 𝑆)
10	8, 9	sylibr 132	1 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆) → 𝑆 = V)

Syntax hints:   → wi 4  ∀wal 1282   = wceq 1284   ∈ wcel 1433  [wsb 1685  ∀wral 2348  Vcvv 2601

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-v 2603

This theorem is referenced by:  setind  4282