Theorem uniex2 7456

Description: The Axiom of Union using the standard abbreviation for union. Given any set 𝑥, its union 𝑦 exists. (Contributed by NM, 4-Jun-2006.)

Assertion

Ref	Expression
uniex2	⊢ ∃𝑦 𝑦 = ∪ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem uniex2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-un 7453	. . . 4 ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		eluni 4833	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥))
3	2	imbi1i 352	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
4	3	albii 1813	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
5	4	exbii 1841	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	1, 5	mpbir 233	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦)
7	6	bm1.3ii 5197	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥)
8		dfcleq 2813	. . 3 ⊢ (𝑦 = ∪ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
9	8	exbii 1841	. 2 ⊢ (∃𝑦 𝑦 = ∪ 𝑥 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
10	7, 9	mpbir 233	1 ⊢ ∃𝑦 𝑦 = ∪ 𝑥

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  ∪ cuni 4830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-uni 4831

This theorem is referenced by:  vuniex  7457