New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  iunn0 GIF version

Theorem iunn0 4026
 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 (x A Bx A 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 2878 . . 3 (x A y y Byx A y B)
2 eliun 3973 . . . 4 (y x A Bx A y B)
32exbii 1582 . . 3 (y y x A Byx A y B)
41, 3bitr4i 243 . 2 (x A y y By y x A B)
5 n0 3559 . . 3 (By y B)
65rexbii 2639 . 2 (x A Bx A y y B)
7 n0 3559 . 2 (x A By y x A B)
84, 6, 73bitr4i 268 1 (x A Bx A B)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∃wex 1541   ∈ wcel 1710   ≠ wne 2516  ∃wrex 2615  ∅c0 3550  ∪ciun 3969 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iun 3971 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator