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Theorem dmsn0el 4887
Description: The domain of a singleton is empty if the singleton's argument contains the empty set. (Contributed by NM, 15-Dec-2008.)
Assertion
Ref Expression
dmsn0el (∅ ∈ 𝐴 → dom {𝐴} = ∅)

Proof of Theorem dmsn0el
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 0nelelxp 4456 . . . . 5 (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
21con2i 592 . . . 4 (∅ ∈ 𝐴 → ¬ 𝐴 ∈ (V × V))
3 dmsnm 4883 . . . 4 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
42, 3sylnib 636 . . 3 (∅ ∈ 𝐴 → ¬ ∃𝑥 𝑥 ∈ dom {𝐴})
5 alnex 1433 . . 3 (∀𝑥 ¬ 𝑥 ∈ dom {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ dom {𝐴})
64, 5sylibr 132 . 2 (∅ ∈ 𝐴 → ∀𝑥 ¬ 𝑥 ∈ dom {𝐴})
7 eq0 3299 . 2 (dom {𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {𝐴})
86, 7sylibr 132 1 (∅ ∈ 𝐴 → dom {𝐴} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1287   = wceq 1289  wex 1426  wcel 1438  Vcvv 2619  c0 3284  {csn 3441   × cxp 4426  dom cdm 4428
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-dm 4438
This theorem is referenced by: (None)
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