Theorem iunn0 4580

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 3225	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		eliun 4524	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3	2	exbii 1774	. . 3 ⊢ (∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
4	1, 3	bitr4i 267	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
5		n0 3931	. . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	5	rexbii 3041	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵)
7		n0 3931	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
8	4, 6, 7	3bitr4i 292	1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅)

Syntax hints:   ↔ wb 196  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  ∅c0 3915  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-nul 3916  df-iun 4522

This theorem is referenced by:  fsuppmapnn0fiubex  12792  lbsextlem2  19159