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Theorem 0domg 9138
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5370, ax-un 7753. (Revised by BTernaryTau, 29-Nov-2024.)
Assertion
Ref Expression
0domg (𝐴𝑉 → ∅ ≼ 𝐴)

Proof of Theorem 0domg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 0ex 5312 . . 3 ∅ ∈ V
2 f1eq1 6799 . . 3 (𝑓 = ∅ → (𝑓:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
3 f10 6881 . . 3 ∅:∅–1-1𝐴
41, 2, 3ceqsexv2d 3532 . 2 𝑓 𝑓:∅–1-1𝐴
5 brdom2g 8994 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1𝐴))
61, 5mpan 690 . 2 (𝐴𝑉 → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1𝐴))
74, 6mpbiri 258 1 (𝐴𝑉 → ∅ ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1775  wcel 2105  Vcvv 3477  c0 4338   class class class wbr 5147  1-1wf1 6559  cdom 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-dom 8985
This theorem is referenced by:  dom0OLD  9141  0sdomg  9142  0sdomgOLD  9143  0dom  9144  sdom0OLD  9147  carddomi2  10007  wdomfil  10098  wdomnumr  10101  hashge0  14422  ufildom1  23949  harn0  43090  safesnsupfidom1o  43406  sn1dom  43515
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