Theorem axpow3 5291

Description: A variant of the Axiom of Power Sets ax-pow 5288. 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 5290	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
2	1	bm1.3ii 5226	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
3		bicom 221	. . . 4 ⊢ ((𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
4	3	albii 1822	. . 3 ⊢ (∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
5	4	exbii 1850	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	2, 5	mpbir 230	1 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)

Syntax hints:   ↔ wb 205  ∀wal 1537  ∃wex 1782   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by: (None)