Theorem axpow3 5329

Description: A variant of the Axiom of Power Sets ax-pow 5326. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)

Assertion

Ref	Expression
axpow3	⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧

Proof of Theorem axpow3

Step	Hyp	Ref	Expression
1		axpow2 5328	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
2	1	sepexi 5255	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
3		bicom1 224	. . 3 ⊢ ((𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥) → (𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦))
4	3	alimi 1843	. 2 ⊢ (∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥) → ∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦))
5	2, 4	eximii 1869	1 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)

This proof depends on syntax axioms:   ↔ wb 209  ∀wal 1567  ∃wex 1811   ⊆ wss 3895

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-sep 5248  ax-pow 5326

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-ss 3896

This theorem is used by: (None)