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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 4568 | . . . . 5 ⊢ (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴) | |
2 | 1 | con2i 616 | . . . 4 ⊢ (∅ ∈ 𝐴 → ¬ 𝐴 ∈ (V × V)) |
3 | dmsnm 5004 | . . . 4 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
4 | 2, 3 | sylnib 665 | . . 3 ⊢ (∅ ∈ 𝐴 → ¬ ∃𝑥 𝑥 ∈ dom {𝐴}) |
5 | alnex 1475 | . . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
6 | 4, 5 | sylibr 133 | . 2 ⊢ (∅ ∈ 𝐴 → ∀𝑥 ¬ 𝑥 ∈ dom {𝐴}) |
7 | eq0 3381 | . 2 ⊢ (dom {𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {𝐴}) | |
8 | 6, 7 | sylibr 133 | 1 ⊢ (∅ ∈ 𝐴 → dom {𝐴} = ∅) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1329 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 ∅c0 3363 {csn 3527 × cxp 4537 dom cdm 4539 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-dm 4549 |
This theorem is referenced by: (None) |
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