Theorem axpow2 3929

Description: A variant of the Axiom of Power Sets ax-pow 3927 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 3927	. 2 ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		dfss2 2934	. . . . 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 1359	. . 3 ⊢ (∀𝑧(𝑧 ⊆ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∀𝑤(𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
5	4	exbii 1496	. 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 1241  ∃wex 1381   ⊆ wss 2917

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  axpow3  3930  pwex  3932