Theorem iunn0 5034

Description: There is a nonempty class in an indexed collection 𝐵(𝑥) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iunn0	⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunn0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 3265	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		eliun 4962	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	exbii 1848	. . 3 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	1, 3	bitr4i 278	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
5		n0 4319	. . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	rexbii 3077	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵)
7		n0 4319	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
8	4, 6, 7	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅)

Syntax hints:   ↔ wb 206  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  ∅c0 4299  ∪ ciun 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-rex 3055  df-v 3452  df-dif 3920  df-nul 4300  df-iun 4960

This theorem is referenced by:  fsuppmapnn0fiubex  13964  lbsextlem2  21076