Theorem axun2 7584

Description: A variant of the Axiom of Union ax-un 7582. 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.)

Assertion

Ref	Expression
axun2	⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))

Distinct variable group:   𝑥,𝑤,𝑦,𝑧

Proof of Theorem axun2

Step	Hyp	Ref	Expression
1		ax-un 7582	. 2 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2	1	bm1.3ii 5230	1 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))

Syntax hints:   ↔ wb 205   ∧ wa 396  ∀wal 1540  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-9 2120  ax-sep 5227  ax-un 7582

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1787

This theorem is referenced by: (None)