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Theorem 0domg 9140
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5365, ax-un 7755. (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 5307 . . 3 ∅ ∈ V
2 f1eq1 6799 . . 3 (𝑓 = ∅ → (𝑓:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
3 f10 6881 . . 3 ∅:∅–1-1𝐴
41, 2, 3ceqsexv2d 3533 . 2 𝑓 𝑓:∅–1-1𝐴
5 brdom2g 8996 . . 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 1779  wcel 2108  Vcvv 3480  c0 4333   class class class wbr 5143  1-1wf1 6558  cdom 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-dom 8987
This theorem is referenced by:  dom0OLD  9143  0sdomg  9144  0sdomgOLD  9145  0dom  9146  sdom0OLD  9149  carddomi2  10010  wdomfil  10101  wdomnumr  10104  hashge0  14426  ufildom1  23934  harn0  43114  safesnsupfidom1o  43430  sn1dom  43539
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