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Theorem iunn0 4964
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 3237 . . 3 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
2 eliun 4898 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32exbii 1849 . . 3 (∃𝑦 𝑦 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
41, 3bitr4i 281 . 2 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
5 n0 4282 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65rexbii 3235 . 2 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝐵)
7 n0 4282 . 2 ( 𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
84, 6, 73bitr4i 306 1 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wex 1781  wcel 2114  wne 3011  wrex 3131  c0 4265   ciun 4894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-nul 4266  df-iun 4896
This theorem is referenced by:  fsuppmapnn0fiubex  13355  lbsextlem2  19922
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