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Mirrors > Home > ILE Home > Th. List > notm0 | GIF version |
Description: A class is not inhabited if and only if it is empty. (Contributed by Jim Kingdon, 1-Jul-2022.) |
Ref | Expression |
---|---|
notm0 | ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eq0 3284 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | alnex 1429 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 1, 2 | bitr2i 183 | 1 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∀wal 1283 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∅c0 3269 |
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 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-v 2614 df-dif 2986 df-nul 3270 |
This theorem is referenced by: mapprc 6337 map0g 6373 exmidfodomrlemim 6728 |
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