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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 3315 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 1443 | . 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | n0rf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
3 | nfcv 2235 | . . . . 5 ⊢ Ⅎ𝑥∅ | |
4 | 2, 3 | cleqf 2259 | . . . 4 ⊢ (𝐴 = ∅ ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
5 | noel 3306 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
6 | 5 | nbn 653 | . . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
7 | 6 | albii 1411 | . . . 4 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ ∅)) |
8 | 4, 7 | bitr4i 186 | . . 3 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
9 | 8 | necon3abii 2298 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥 ∈ 𝐴) |
10 | 1, 9 | sylibr 133 | 1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∀wal 1294 = wceq 1296 ∃wex 1433 ∈ wcel 1445 Ⅎwnfc 2222 ≠ wne 2262 ∅c0 3302 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-v 2635 df-dif 3015 df-nul 3303 |
This theorem is referenced by: n0r 3315 abn0r 3326 |
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