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Mirrors > Home > ILE Home > Th. List > n0rf | GIF version |
Description: An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class 𝐴 nonempty if 𝐴 ≠ ∅ 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 3234 requires only that 𝑥 not be free in, rather than not occur in, 𝐴. (Contributed by Jim Kingdon, 31-Jul-2018.) |
Ref | Expression |
---|---|
n0rf.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
n0rf | ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exalim 1391 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | n0rf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑥∅ | |
4 | 2, 3 | cleqf 2201 | . . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
5 | noel 3228 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | 5 | nbn 615 | . . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
7 | 6 | albii 1359 | . . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
8 | 4, 7 | bitr4i 176 | . . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
9 | 8 | necon3abii 2241 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
10 | 1, 9 | sylibr 137 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 98 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Ⅎwnfc 2165 ≠ wne 2204 ∅c0 3224 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-nul 3225 |
This theorem is referenced by: n0r 3234 abn0r 3243 |
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