Theorem setindel 4555

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 2294	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆)
2	1	ralbii 2496	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝑆)
3		df-ral 2473	. . . . . 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 1481	. . 3 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝑥 ∈ 𝑆 → 𝑥 ∈ 𝑆) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑆) → 𝑥 ∈ 𝑆))
7		ax-setind 4554	. . 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 3457	. 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 1773   ∈ wcel 2160  ∀wral 2468  Vcvv 2752

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-v 2754

This theorem is referenced by:  setind  4556