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 4537 | . . . 4 ⊢ ¬ ∅ ∈ (V × V) | |
2 | dmsnm 4974 | . . . 4 ⊢ (∅ ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {∅}) | |
3 | 1, 2 | mtbi 644 | . . 3 ⊢ ¬ ∃𝑥 𝑥 ∈ dom {∅} |
4 | alnex 1460 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {∅} ↔ ¬ ∃𝑥 𝑥 ∈ dom {∅}) | |
5 | 3, 4 | mpbir 145 | . 2 ⊢ ∀𝑥 ¬ 𝑥 ∈ dom {∅} |
6 | eq0 3351 | . 2 ⊢ (dom {∅} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {∅}) | |
7 | 5, 6 | mpbir 145 | 1 ⊢ dom {∅} = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∀wal 1314 = wceq 1316 ∃wex 1453 ∈ wcel 1465 Vcvv 2660 ∅c0 3333 {csn 3497 × cxp 4507 dom cdm 4509 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-dm 4519 |
This theorem is referenced by: cnvsn0 4977 1st0 6010 2nd0 6011 |
Copyright terms: Public domain | W3C validator |