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Mirrors > Home > NFE Home > Th. List > iunn0 | Unicode version |
Description: There is a non-empty class in an indexed collection iff the indexed union of them is non-empty. (Contributed by NM, 15-Oct-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
iunn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2878 | . . 3 | |
2 | eliun 3973 | . . . 4 | |
3 | 2 | exbii 1582 | . . 3 |
4 | 1, 3 | bitr4i 243 | . 2 |
5 | n0 3559 | . . 3 | |
6 | 5 | rexbii 2639 | . 2 |
7 | n0 3559 | . 2 | |
8 | 4, 6, 7 | 3bitr4i 268 | 1 |
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-1 5 ax-2 6 ax-3 7 ax-mp 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) |
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