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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 nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.) |
Ref | Expression |
---|---|
iin0r |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4142 |
. . . . 5
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2 | n0i 3440 |
. . . . 5
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3 | 1, 2 | ax-mp 5 |
. . . 4
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4 | 0iin 3957 |
. . . . 5
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5 | 4 | eqeq1i 2195 |
. . . 4
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6 | 3, 5 | mtbir 672 |
. . 3
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7 | iineq1 3912 |
. . . 4
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8 | 7 | eqeq1d 2196 |
. . 3
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9 | 6, 8 | mtbiri 676 |
. 2
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10 | 9 | necon2ai 2411 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-nul 4141 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-v 2751 df-dif 3143 df-nul 3435 df-iin 3901 |
This theorem is referenced by: (None) |
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