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Theorem dmsn0el 4733
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 4316 . . . . 5 (A (V × V) → ¬ ∅ A)
21con2i 557 . . . 4 (∅ A → ¬ A (V × V))
3 dmsnm 4729 . . . 4 (A (V × V) ↔ x x dom {A})
42, 3sylnib 600 . . 3 (∅ A → ¬ x x dom {A})
5 alnex 1385 . . 3 (x ¬ x dom {A} ↔ ¬ x x dom {A})
64, 5sylibr 137 . 2 (∅ Ax ¬ x dom {A})
7 eq0 3233 . 2 (dom {A} = ∅ ↔ x ¬ x dom {A})
86, 7sylibr 137 1 (∅ A → dom {A} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wal 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  c0 3218  {csn 3367   × cxp 4286  dom cdm 4288
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-dm 4298
This theorem is referenced by: (None)
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