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Theorem 0dom 9101
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
0sdom.1 𝐴 ∈ V
Assertion
Ref Expression
0dom ∅ ≼ 𝐴

Proof of Theorem 0dom
StepHypRef Expression
1 0sdom.1 . 2 𝐴 ∈ V
2 0domg 9095 . 2 (𝐴 ∈ V → ∅ ≼ 𝐴)
31, 2ax-mp 5 1 ∅ ≼ 𝐴
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3466  c0 4314   class class class wbr 5138  cdom 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-dom 8936
This theorem is referenced by:  domunsn  9122  mapdom1  9137  mapdom2  9143  fodomfi  9320  marypha1lem  9423  card2inf  9545  iunfictbso  10104  konigthlem  10558  cctop  22819  ovol0  25332  fvconstdomi  47680
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