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Mirrors > Home > ILE Home > Th. List > axpow3 | GIF version |
Description: A variant of the Axiom of Power Sets ax-pow 3918. 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.) |
Ref | Expression |
---|---|
axpow3 | ⊢ ∃y∀z(z ⊆ x ↔ z ∈ y) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axpow2 3920 | . . 3 ⊢ ∃y∀z(z ⊆ x → z ∈ y) | |
2 | 1 | bm1.3ii 3869 | . 2 ⊢ ∃y∀z(z ∈ y ↔ z ⊆ x) |
3 | bicom 128 | . . . 4 ⊢ ((z ⊆ x ↔ z ∈ y) ↔ (z ∈ y ↔ z ⊆ x)) | |
4 | 3 | albii 1356 | . . 3 ⊢ (∀z(z ⊆ x ↔ z ∈ y) ↔ ∀z(z ∈ y ↔ z ⊆ x)) |
5 | 4 | exbii 1493 | . 2 ⊢ (∃y∀z(z ⊆ x ↔ z ∈ y) ↔ ∃y∀z(z ∈ y ↔ z ⊆ x)) |
6 | 2, 5 | mpbir 134 | 1 ⊢ ∃y∀z(z ⊆ x ↔ z ∈ y) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀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-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 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: (None) |
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