Theorem iunn0 4981

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 3249	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		eliun 4915	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	exbii 1844	. . 3 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	1, 3	bitr4i 280	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
5		n0 4309	. . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	rexbii 3247	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵)
7		n0 4309	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
8	4, 6, 7	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅)

Syntax hints:   ↔ wb 208  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016  ∃wrex 3139  ∅c0 4290  ∪ ciun 4911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3496  df-dif 3938  df-nul 4291  df-iun 4913

This theorem is referenced by:  fsuppmapnn0fiubex  13354  lbsextlem2  19925