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Theorem tr0 3865
 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 3255 . 2 ∅ ⊆ 𝒫 ∅
2 dftr4 3859 . 2 (Tr ∅ ↔ ∅ ⊆ 𝒫 ∅)
31, 2mpbir 134 1 Tr ∅
 Colors of variables: wff set class Syntax hints:   ⊆ wss 2917  ∅c0 3224  𝒫 cpw 3359  Tr wtr 3854 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-uni 3581  df-tr 3855 This theorem is referenced by:  ord0  4128  ordom  4329
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