MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0domg Structured version   Visualization version   GIF version

Theorem 0domg 9097
Description: Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5354, ax-un 7719. (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 5298 . . 3 ∅ ∈ V
2 f1eq1 6773 . . 3 (𝑓 = ∅ → (𝑓:∅–1-1𝐴 ↔ ∅:∅–1-1𝐴))
3 f10 6857 . . 3 ∅:∅–1-1𝐴
41, 2, 3ceqsexv2d 3521 . 2 𝑓 𝑓:∅–1-1𝐴
5 brdom2g 8948 . . 3 ((∅ ∈ V ∧ 𝐴𝑉) → (∅ ≼ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1𝐴))
61, 5mpan 687 . 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 1773  wcel 2098  Vcvv 3466  c0 4315   class class class wbr 5139  1-1wf1 6531  cdom 8934
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 5290  ax-nul 5297  ax-pr 5418
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 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-dom 8938
This theorem is referenced by:  dom0OLD  9100  0sdomg  9101  0sdomgOLD  9102  0dom  9103  sdom0OLD  9106  carddomi2  9962  wdomfil  10053  wdomnumr  10056  hashge0  14348  ufildom1  23774  harn0  42394  safesnsupfidom1o  42717  sn1dom  42826
  Copyright terms: Public domain W3C validator