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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 3453 | . 2 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴) | |
2 | alnex 1509 | . 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴) | |
3 | 1, 2 | bitr2i 185 | 1 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 ∀wal 1361 = wceq 1363 ∃wex 1502 ∈ wcel 2158 ∅c0 3434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-dif 3143 df-nul 3435 |
This theorem is referenced by: disjnim 4006 pwntru 4211 exmidn0m 4213 mapprc 6666 map0g 6702 ixpprc 6733 ixp0 6745 exmidfodomrlemim 7214 ntreq0 13928 blssioo 14341 pwtrufal 15044 |
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