Theorem uniex2 4501

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 variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zfun 4499	. . . 4 ⊢ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		eluni 3867	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))
3	2	imbi1i 238	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
4	3	albii 1494	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
5	4	exbii 1629	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	1, 5	mpbir 146	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦)
7	6	bm1.3ii 4181	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥)
8		dfcleq 2201	. . 3 ⊢ (𝑦 = ∪ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
9	8	exbii 1629	. 2 ⊢ (∃𝑦 𝑦 = ∪ 𝑥 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
10	7, 9	mpbir 146	1 ⊢ ∃𝑦 𝑦 = ∪ 𝑥

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2178  ∪ cuni 3864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-uni 3865

This theorem is referenced by:  uniex  4502