Theorem axpow3 4101

Description: A variant of the Axiom of Power Sets ax-pow 4098. 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 4100	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦)
2	1	bm1.3ii 4049	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥)
3		bicom 139	. . . 4 ⊢ ((𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
4	3	albii 1446	. . 3 ⊢ (∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
5	4	exbii 1584	. 2 ⊢ (∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ⊆ 𝑥))
6	2, 5	mpbir 145	1 ⊢ ∃𝑦∀𝑧(𝑧 ⊆ 𝑥 ↔ 𝑧 ∈ 𝑦)

Syntax hints:   ↔ wb 104  ∀wal 1329  ∃wex 1468   ⊆ wss 3071

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-in 3077  df-ss 3084

This theorem is referenced by: (None)