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

Theorem unisuc 4088
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 3081 . 2 ( AA ↔ ( AA) = A)
2 df-tr 3818 . 2 (Tr A AA)
3 df-suc 4046 . . . . 5 suc A = (A ∪ {A})
43unieqi 3553 . . . 4 suc A = (A ∪ {A})
5 uniun 3562 . . . 4 (A ∪ {A}) = ( A {A})
6 unisuc.1 . . . . . 6 A V
76unisn 3559 . . . . 5 {A} = A
87uneq2i 3062 . . . 4 ( A {A}) = ( AA)
94, 5, 83eqtri 2037 . . 3 suc A = ( AA)
109eqeq1i 2020 . 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 1223   wcel 1366  Vcvv 2526  cun 2883  wss 2885  {csn 3339   cuni 3543  Tr wtr 3817  suc csuc 4040
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-sn 3345  df-pr 3346  df-uni 3544  df-tr 3818  df-suc 4046
This theorem is referenced by:  onunisuci  4107  ordsucunielexmid  4188  tfrexlem  5858
  Copyright terms: Public domain W3C validator