Theorem setind 4264

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 2934	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
2	1	imbi1i 227	. . 3 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
3	2	albii 1359	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
4		setindel 4263	. 2 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) → 𝐴 = V)
5	3, 4	sylbi 114	1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931

This theorem is referenced by:  setind2  4265