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Mirrors > Home > ILE Home > Th. List > n0rf | Unicode version |
Description: An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class nonempty if and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3427 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.) |
Ref | Expression |
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n0rf.1 |
Ref | Expression |
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n0rf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exalim 1495 | . 2 | |
2 | n0rf.1 | . . . . 5 | |
3 | nfcv 2312 | . . . . 5 | |
4 | 2, 3 | cleqf 2337 | . . . 4 |
5 | noel 3418 | . . . . . 6 | |
6 | 5 | nbn 694 | . . . . 5 |
7 | 6 | albii 1463 | . . . 4 |
8 | 4, 7 | bitr4i 186 | . . 3 |
9 | 8 | necon3abii 2376 | . 2 |
10 | 1, 9 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 104 wal 1346 wceq 1348 wex 1485 wcel 2141 wnfc 2299 wne 2340 c0 3414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-dif 3123 df-nul 3415 |
This theorem is referenced by: n0r 3427 abn0r 3438 |
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