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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 3381 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | alnex 1475 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 1, 2 | bitr2i 184 | 1 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∀wal 1329 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∅c0 3363 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-dif 3073 df-nul 3364 |
This theorem is referenced by: disjnim 3920 pwntru 4122 exmidn0m 4124 mapprc 6546 map0g 6582 ixpprc 6613 ixp0 6625 exmidfodomrlemim 7057 ntreq0 12301 blssioo 12714 pwtrufal 13192 |
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