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Theorem unisucg 4100
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 (A 𝑉 → (Tr A suc A = A))

Proof of Theorem unisucg
StepHypRef Expression
1 df-suc 4057 . . . . . 6 suc A = (A ∪ {A})
21unieqi 3564 . . . . 5 suc A = (A ∪ {A})
3 uniun 3573 . . . . 5 (A ∪ {A}) = ( A {A})
42, 3eqtri 2042 . . . 4 suc A = ( A {A})
5 unisng 3571 . . . . 5 (A 𝑉 {A} = A)
65uneq2d 3074 . . . 4 (A 𝑉 → ( A {A}) = ( AA))
74, 6syl5eq 2066 . . 3 (A 𝑉 suc A = ( AA))
87eqeq1d 2030 . 2 (A 𝑉 → ( suc A = A ↔ ( AA) = A))
9 df-tr 3829 . . 3 (Tr A AA)
10 ssequn1 3090 . . 3 ( AA ↔ ( AA) = A)
119, 10bitri 173 . 2 (Tr A ↔ ( AA) = A)
128, 11syl6rbbr 188 1 (A 𝑉 → (Tr A suc A = A))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  cun 2892  wss 2894  {csn 3350   cuni 3554  Tr wtr 3828  suc csuc 4051
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-pr 3357  df-uni 3555  df-tr 3829  df-suc 4057
This theorem is referenced by:  nlimsucg  4226
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