Theorem setind 4608

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		ssalel 3192	. . . 4 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
2	1	imbi1i 238	. . 3 ⊢ ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
3	2	albii 1496	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
4		setindel 4607	. 2 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴) → 𝐴 = V)
5	3, 4	sylbi 121	1 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)

Syntax hints:   → wi 4  ∀wal 1373   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ⊆ wss 3177

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-ral 2493  df-v 2781  df-in 3183  df-ss 3190

This theorem is referenced by:  setind2  4609