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Theorem unisuc 4150
 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
Assertion
Ref Expression
unisuc

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 3113 . 2
2 df-tr 3855 . 2
3 df-suc 4108 . . . . 5
43unieqi 3590 . . . 4
5 uniun 3599 . . . 4
6 unisuc.1 . . . . . 6
76unisn 3596 . . . . 5
87uneq2i 3094 . . . 4
94, 5, 83eqtri 2064 . . 3
109eqeq1i 2047 . 2
111, 2, 103bitr4i 201 1
 Colors of variables: wff set class Syntax hints:   wb 98   wceq 1243   wcel 1393  cvv 2557   cun 2915   wss 2917  csn 3375  cuni 3580   wtr 3854   csuc 4102 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 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-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-suc 4108 This theorem is referenced by:  onunisuci  4169  ordsucunielexmid  4256  tfrexlem  5948
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