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Theorem dm0 4753
Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dm0 dom ∅ = ∅

Proof of Theorem dm0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3381 . 2 (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2 noel 3367 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
32nex 1476 . . 3 ¬ ∃𝑦𝑥, 𝑦⟩ ∈ ∅
4 vex 2689 . . . 4 𝑥 ∈ V
54eldm2 4737 . . 3 (𝑥 ∈ dom ∅ ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ∅)
63, 5mtbir 660 . 2 ¬ 𝑥 ∈ dom ∅
71, 6mpgbir 1429 1 dom ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1331  wex 1468  wcel 1480  c0 3363  cop 3530  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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
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-v 2688  df-dif 3073  df-un 3075  df-nul 3364  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-dm 4549
This theorem is referenced by:  rn0  4795  sqxpeq0  4962  fn0  5242  f0dom0  5316  f1o00  5402  rdg0  6284  frec0g  6294  ennnfonelemj0  11925  ennnfonelem1  11931  ennnfonelemkh  11936  ennnfonelemhf1o  11937
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