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Theorem iunn0 5071
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 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 𝐵 ≠ ∅)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunn0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rexcom4 3285 . . 3 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
2 eliun 4999 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32exbii 1844 . . 3 (∃𝑦 𝑦 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
41, 3bitr4i 278 . 2 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
5 n0 4358 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65rexbii 3091 . 2 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝐵)
7 n0 4358 . 2 ( 𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
84, 6, 73bitr4i 303 1 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 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
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