ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tr0 GIF version

Theorem tr0 4130
Description: The empty set is transitive. (Contributed by NM, 16-Sep-1993.)
Assertion
Ref Expression
tr0 Tr ∅

Proof of Theorem tr0
StepHypRef Expression
1 0ss 3476 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 4124 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 146 1 Tr ∅
Colors of variables: wff set class
Syntax hints:  wss 3144  c0 3437  𝒫 cpw 3593  Tr wtr 4119
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-uni 3828  df-tr 4120
This theorem is referenced by:  ord0  4412  ordom  4627
  Copyright terms: Public domain W3C validator