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Theorem dmsn0el 4817
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 4400 . . . . 5 (𝐴 ∈ (V × V) → ¬ ∅ ∈ 𝐴)
21con2i 567 . . . 4 (∅ ∈ 𝐴 → ¬ 𝐴 ∈ (V × V))
3 dmsnm 4813 . . . 4 (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴})
42, 3sylnib 611 . . 3 (∅ ∈ 𝐴 → ¬ ∃𝑥 𝑥 ∈ dom {𝐴})
5 alnex 1404 . . 3 (∀𝑥 ¬ 𝑥 ∈ dom {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ dom {𝐴})
64, 5sylibr 141 . 2 (∅ ∈ 𝐴 → ∀𝑥 ¬ 𝑥 ∈ dom {𝐴})
7 eq0 3266 . 2 (dom {𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom {𝐴})
86, 7sylibr 141 1 (∅ ∈ 𝐴 → dom {𝐴} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  c0 3251  {csn 3402   × cxp 4370  dom cdm 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-dm 4382
This theorem is referenced by: (None)
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