Theorem axpow2 3920

Description: A variant of the Axiom of Power Sets ax-pow 3918 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 3918	. 2 ⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y)
2		dfss2 2928	. . . . 5 ⊢ (z ⊆ x ↔ ∀w(w ∈ z → w ∈ x))
3	2	imbi1i 227	. . . 4 ⊢ ((z ⊆ x → z ∈ y) ↔ (∀w(w ∈ z → w ∈ x) → z ∈ y))
4	3	albii 1356	. . 3 ⊢ (∀z(z ⊆ x → z ∈ y) ↔ ∀z(∀w(w ∈ z → w ∈ x) → z ∈ y))
5	4	exbii 1493	. 2 ⊢ (∃y∀z(z ⊆ x → z ∈ y) ↔ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y))
6	1, 5	mpbir 134	1 ⊢ ∃y∀z(z ⊆ x → z ∈ y)

Syntax hints:   → wi 4  ∀wal 1240  ∃wex 1378   ⊆ wss 2911

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  axpow3  3921  pwex  3923