Theorem setind 4370

Description: Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)

Assertion

Ref	Expression
setind	⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

Distinct variable group:   𝑥,𝐴

Proof of Theorem setind

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss2 3017	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
2	1	imbi1i 237	. . 3 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
3	2	albii 1405	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
4		setindel 4369	. 2 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) → 𝐴 = V)
5	3, 4	sylbi 120	1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1288   = wceq 1290   ∈ wcel 1439  Vcvv 2622   ⊆ wss 3002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-setind 4368

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-v 2624  df-in 3008  df-ss 3015

This theorem is referenced by:  setind2  4371