MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1o0 Structured version   Visualization version   GIF version

Theorem f1o0 6870
Description: One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
Assertion
Ref Expression
f1o0 ∅:∅–1-1-onto→∅

Proof of Theorem f1o0
StepHypRef Expression
1 eqid 2726 . 2 ∅ = ∅
2 f1o00 6868 . 2 (∅:∅–1-1-onto→∅ ↔ (∅ = ∅ ∧ ∅ = ∅))
31, 1, 2mpbir2an 709 1 ∅:∅–1-1-onto→∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  c0 4323  1-1-ontowf1o 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-mo 2529  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3465  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4324  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551
This theorem is referenced by:  en0  9039  en0OLD  9040  en0r  9042  brwdom2  9607  cnfcom  9734  ackbij2lem2  10272  eupth0  30142  f1ocnt  32705  1arithidom  33416  iso0  44016
  Copyright terms: Public domain W3C validator