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Theorem unisucg 4373
 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 Jim Kingdon, 18-Aug-2019.)
Assertion
Ref Expression
unisucg

Proof of Theorem unisucg
StepHypRef Expression
1 df-tr 4063 . . 3
2 ssequn1 3277 . . 3
31, 2bitri 183 . 2
4 df-suc 4330 . . . . . 6
54unieqi 3782 . . . . 5
6 uniun 3791 . . . . 5
75, 6eqtri 2178 . . . 4
8 unisng 3789 . . . . 5
98uneq2d 3261 . . . 4
107, 9syl5eq 2202 . . 3
1110eqeq1d 2166 . 2
123, 11bitr4id 198 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1335   wcel 2128   cun 3100   wss 3102  csn 3560  cuni 3772   wtr 4062   csuc 4324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-uni 3773  df-tr 4063  df-suc 4330 This theorem is referenced by:  onsucuni2  4521  nlimsucg  4523  ctmlemr  7042  nnsf  13538  peano4nninf  13539
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