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Mirrors > Home > MPE Home > Th. List > iunn0 | Structured version Visualization version Unicode version |
Description: There is a nonempty class in an indexed collection iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iunn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 3225 | . . 3 | |
2 | eliun 4524 | . . . 4 | |
3 | 2 | exbii 1774 | . . 3 |
4 | 1, 3 | bitr4i 267 | . 2 |
5 | n0 3931 | . . 3 | |
6 | 5 | rexbii 3041 | . 2 |
7 | n0 3931 | . 2 | |
8 | 4, 6, 7 | 3bitr4i 292 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wex 1704 wcel 1990 wne 2794 wrex 2913 c0 3915 ciun 4520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-dif 3577 df-nul 3916 df-iun 4522 |
This theorem is referenced by: fsuppmapnn0fiubex 12792 lbsextlem2 19159 |
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