![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > dmsn0 | GIF version |
Description: The domain of the singleton of the empty set is empty. (Contributed by NM, 30-Jan-2004.) |
Ref | Expression |
---|---|
dmsn0 | ⊢ dom {∅} = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 4428 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
2 | dmsnm 4850 | . . . 4 ⊢ (∅ ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {∅}) | |
3 | 1, 2 | mtbi 628 | . . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ dom {∅} |
4 | alnex 1429 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {∅} ↔ ¬ ∃𝑥 𝑥 ∈ dom {∅}) | |
5 | 3, 4 | mpbir 144 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ dom {∅} |
6 | eq0 3284 | . 2 ⊢ (dom {∅} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {∅}) | |
7 | 5, 6 | mpbir 144 | 1 ⊢ dom {∅} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1283 = wceq 1285 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 ∅c0 3269 {csn 3422 × cxp 4399 dom cdm 4401 |
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-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 |
This theorem depends on definitions: df-bi 115 df-3an 922 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-ne 2250 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-xp 4407 df-dm 4411 |
This theorem is referenced by: cnvsn0 4853 1st0 5850 2nd0 5851 |
Copyright terms: Public domain | W3C validator |