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Mirrors > Home > ILE Home > Th. List > axpow2 | GIF version |
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.) |
Ref | Expression |
---|---|
axpow2 | ⊢ ∃y∀z(z ⊆ x → z ∈ y) |
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) |
Colors of variables: wff set class |
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 |
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