MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunn0 Structured version   Visualization version   GIF version

Theorem iunn0 4510
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 3197 . . 3 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
2 eliun 4454 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴 𝑦𝐵)
32exbii 1763 . . 3 (∃𝑦 𝑦 𝑥𝐴 𝐵 ↔ ∃𝑦𝑥𝐴 𝑦𝐵)
41, 3bitr4i 265 . 2 (∃𝑥𝐴𝑦 𝑦𝐵 ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
5 n0 3889 . . 3 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
65rexbii 3022 . 2 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑥𝐴𝑦 𝑦𝐵)
7 n0 3889 . 2 ( 𝑥𝐴 𝐵 ≠ ∅ ↔ ∃𝑦 𝑦 𝑥𝐴 𝐵)
84, 6, 73bitr4i 290 1 (∃𝑥𝐴 𝐵 ≠ ∅ ↔ 𝑥𝐴 𝐵 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wex 1694  wcel 1976  wne 2779  wrex 2896  c0 3873   ciun 4449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-nul 3874  df-iun 4451
This theorem is referenced by:  fsuppmapnn0fiubex  12609  lbsextlem2  18926
  Copyright terms: Public domain W3C validator