Theorem uniex2 4366

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 4364	. . . 4 ⊢ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
2		eluni 3747	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))
3	2	imbi1i 237	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
4	3	albii 1447	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
5	4	exbii 1585	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	1, 5	mpbir 145	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦)
7	6	bm1.3ii 4057	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥)
8		dfcleq 2134	. . 3 ⊢ (𝑦 = ∪ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
9	8	exbii 1585	. 2 ⊢ (∃𝑦 𝑦 = ∪ 𝑥 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
10	7, 9	mpbir 145	1 ⊢ ∃𝑦 𝑦 = ∪ 𝑥

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∪ cuni 3744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-uni 3745

This theorem is referenced by:  uniex  4367