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Mirrors > Home > ILE Home > Th. List > axun2 | GIF version |
Description: A variant of the Axiom of Union ax-un 4188. For any set 𝑥, there exists a set 𝑦 whose members are exactly the members of the members of 𝑥 i.e. the union of 𝑥. Axiom Union of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.) |
Ref | Expression |
---|---|
axun2 | ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-un 4188 | . 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦) | |
2 | 1 | bm1.3ii 3899 | 1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∀wal 1282 ∃wex 1421 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-sep 3896 ax-un 4188 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: (None) |
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