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

Theorem unisuc 4097
 Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisuc.1 A V
Assertion
Ref Expression
unisuc (Tr A suc A = A)

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 3089 . 2 ( AA ↔ ( AA) = A)
2 df-tr 3828 . 2 (Tr A AA)
3 df-suc 4055 . . . . 5 suc A = (A ∪ {A})
43unieqi 3563 . . . 4 suc A = (A ∪ {A})
5 uniun 3572 . . . 4 (A ∪ {A}) = ( A {A})
6 unisuc.1 . . . . . 6 A V
76unisn 3569 . . . . 5 {A} = A
87uneq2i 3070 . . . 4 ( A {A}) = ( AA)
94, 5, 83eqtri 2047 . . 3 suc A = ( AA)
109eqeq1i 2030 . 2 ( suc A = A ↔ ( AA) = A)
111, 2, 103bitr4i 201 1 (Tr A suc A = A)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1375  Vcvv 2534   ∪ cun 2891   ⊆ wss 2893  {csn 3349  ∪ cuni 3553  Tr wtr 3827  suc csuc 4049 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-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-sn 3355  df-pr 3356  df-uni 3554  df-tr 3828  df-suc 4055 This theorem is referenced by:  onunisuci  4117  ordsucunielexmid  4198  tfrexlem  5865
 Copyright terms: Public domain W3C validator