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Mirrors > Home > ILE Home > Th. List > iin0r | Unicode version |
Description: If an indexed intersection of the empty set is empty, the index set is non-empty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Ref | Expression |
---|---|
iin0r |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 3905 | . . . . 5 | |
2 | n0i 3256 | . . . . 5 | |
3 | 1, 2 | ax-mp 7 | . . . 4 |
4 | 0iin 3736 | . . . . 5 | |
5 | 4 | eqeq1i 2088 | . . . 4 |
6 | 3, 5 | mtbir 628 | . . 3 |
7 | iineq1 3692 | . . . 4 | |
8 | 7 | eqeq1d 2089 | . . 3 |
9 | 6, 8 | mtbiri 632 | . 2 |
10 | 9 | necon2ai 2299 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wceq 1284 wcel 1433 wne 2245 cvv 2601 c0 3251 ciin 3679 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-nul 3904 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-v 2603 df-dif 2975 df-nul 3252 df-iin 3681 |
This theorem is referenced by: (None) |
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