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Theorem 0domg 9051
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5325, ax-un 7677. (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 5269 . . 3 ∅ ∈ V
2 f1eq1 6738 . . 3 (𝑓 = ∅ → (𝑓:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
3 f10 6822 . . 3 ∅:∅–1-1𝐴
41, 2, 3ceqsexv2d 3500 . 2 𝑓 𝑓:∅–1-1𝐴
5 brdom2g 8902 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1𝐴))
61, 5mpan 689 . 2 (𝐴𝑉 → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1𝐴))
74, 6mpbiri 258 1 (𝐴𝑉 → ∅ ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wex 1782  wcel 2107  Vcvv 3448  c0 4287   class class class wbr 5110  1-1wf1 6498  cdom 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-dom 8892
This theorem is referenced by:  dom0OLD  9054  0sdomg  9055  0sdomgOLD  9056  0dom  9057  sdom0OLD  9060  carddomi2  9913  wdomfil  10004  wdomnumr  10007  hashge0  14294  ufildom1  23293  harn0  41458  safesnsupfidom1o  41763  sn1dom  41872
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