ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iso0 GIF version

Theorem iso0 5821
Description: The empty set is an 𝑅, 𝑆 isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
Assertion
Ref Expression
iso0 ∅ Isom 𝑅, 𝑆 (∅, ∅)

Proof of Theorem iso0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1o0 5500 . 2 ∅:∅–1-1-onto→∅
2 ral0 3526 . 2 𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))
3 df-isom 5227 . 2 (∅ Isom 𝑅, 𝑆 (∅, ∅) ↔ (∅:∅–1-1-onto→∅ ∧ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥𝑅𝑦 ↔ (∅‘𝑥)𝑆(∅‘𝑦))))
41, 2, 3mpbir2an 942 1 ∅ Isom 𝑅, 𝑆 (∅, ∅)
Colors of variables: wff set class
Syntax hints:  wb 105  wral 2455  c0 3424   class class class wbr 4005  1-1-ontowf1o 5217  cfv 5218   Isom wiso 5219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-isom 5227
This theorem is referenced by:  zfz1iso  10824
  Copyright terms: Public domain W3C validator