Theorem axpow3 4261

Description: A variant of the Axiom of Power Sets ax-pow 4258. 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 4260	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
2	1	bm1.3ii 4205	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
3		bicom 140	. . . 4 ⊢ ((𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
4	3	albii 1516	. . 3 ⊢ (∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
5	4	exbii 1651	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	2, 5	mpbir 146	1 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)

Syntax hints:   ↔ wb 105  ∀wal 1393  ∃wex 1538   ⊆ wss 3197

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-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210

This theorem is referenced by: (None)