Theorem setind 4575

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 3172	. . . 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 1484	. 2 ⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
4		setindel 4574	. 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 1362   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ⊆ wss 3157

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170

This theorem is referenced by:  setind2  4576