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Theorem iunn0 4688
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 3329 . . 3 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
2 eliun 4632 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32exbii 1887 . . 3 (∃𝑦 𝑦 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
41, 3bitr4i 267 . 2 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
5 n0 4039 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65rexbii 3143 . 2 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝐵)
7 n0 4039 . 2 ( 𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
84, 6, 73bitr4i 292 1 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1817  wcel 2103  wne 2896  wrex 3015  c0 4023   ciun 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-v 3306  df-dif 3683  df-nul 4024  df-iun 4630
This theorem is referenced by:  fsuppmapnn0fiubex  12907  lbsextlem2  19282
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