MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fun0 Structured version   Visualization version   GIF version

Theorem fun0 6191
Description: The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
Assertion
Ref Expression
fun0 Fun ∅

Proof of Theorem fun0
StepHypRef Expression
1 0ss 4199 . 2 ∅ ⊆ {⟨∅, ∅⟩}
2 0ex 5016 . . 3 ∅ ∈ V
32, 2funsn 6179 . 2 Fun {⟨∅, ∅⟩}
4 funss 6146 . 2 (∅ ⊆ {⟨∅, ∅⟩} → (Fun {⟨∅, ∅⟩} → Fun ∅))
51, 3, 4mp2 9 1 Fun ∅
Colors of variables: wff setvar class
Syntax hints:  wss 3798  c0 4146  {csn 4399  cop 4405  Fun wfun 6121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-fun 6129
This theorem is referenced by:  funcnv0  6192  fn0  6248  f10  6414  0fsupp  8572  strle1  16339  lubfun  17340  glbfun  17353  1pthdlem1  27507
  Copyright terms: Public domain W3C validator