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Mirrors > Home > ILE Home > Th. List > dmsn0el | GIF version |
Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.) |
Ref | Expression |
---|---|
dmsn0el | ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelelxp 4391 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴) | |
2 | 1 | con2i 589 | . . . 4 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 ∈ (V × V)) |
3 | dmsnm 4806 | . . . 4 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
4 | 2, 3 | sylnib 633 | . . 3 ⊢ (∅ ∈ 𝐴 → ¬ ∃𝑥 𝑥 ∈ dom {𝐴}) |
5 | alnex 1428 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
6 | 4, 5 | sylibr 132 | . 2 ⊢ (∅ ∈ 𝐴 → ∀𝑥 ¬ 𝑥 ∈ dom {𝐴}) |
7 | eq0 3266 | . 2 ⊢ (dom {𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {𝐴}) | |
8 | 6, 7 | sylibr 132 | 1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1282 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 ∅c0 3251 {csn 3398 × cxp 4361 dom cdm 4363 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-dm 4373 |
This theorem is referenced by: (None) |
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