Theorem axpow2 5385

Description: A variant of the Axiom of Power Sets ax-pow 5383 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Assertion

Ref	Expression
axpow2	⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axpow2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-pow 5383	. 2 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		df-ss 3993	. . . . 5 ⊢ (𝑧 ⊆ 𝑥 ↔ ∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥))
3	2	imbi1i 349	. . . 4 ⊢ ((𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
4	3	albii 1817	. . 3 ⊢ (∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
5	4	exbii 1846	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	1, 5	mpbir 231	1 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1777   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-ex 1778  df-ss 3993

This theorem is referenced by:  axpow3  5386  vpwex  5395