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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 4373 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴) | |
2 | 1 | con2i 557 | . . . 4 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 ∈ (V × V)) |
3 | dmsnm 4786 | . . . 4 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
4 | 2, 3 | sylnib 601 | . . 3 ⊢ (∅ ∈ 𝐴 → ¬ ∃𝑥 𝑥 ∈ dom {𝐴}) |
5 | alnex 1388 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
6 | 4, 5 | sylibr 137 | . 2 ⊢ (∅ ∈ 𝐴 → ∀𝑥 ¬ 𝑥 ∈ dom {𝐴}) |
7 | eq0 3239 | . 2 ⊢ (dom {𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {𝐴}) | |
8 | 6, 7 | sylibr 137 | 1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1241 = wceq 1243 ∃wex 1381 ∈ wcel 1393 Vcvv 2557 ∅c0 3224 {csn 3375 × cxp 4343 dom cdm 4345 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-dm 4355 |
This theorem is referenced by: (None) |
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