![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iunn0 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
iunn0 | ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3285 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) | |
2 | eliun 4999 | . . . 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 278 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
5 | n0 4358 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐵) | |
6 | 5 | rexbii 3091 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 ∈ 𝐵) |
7 | n0 4358 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) | |
8 | 4, 6, 7 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ≠ ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 ∅c0 4338 ∪ ciun 4995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-rex 3068 df-v 3479 df-dif 3965 df-nul 4339 df-iun 4997 |
This theorem is referenced by: fsuppmapnn0fiubex 14029 lbsextlem2 21178 |
Copyright terms: Public domain | W3C validator |