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Theorem unisucg 4117
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 4074 . . . . . 6 suc A = (A ∪ {A})
21unieqi 3581 . . . . 5 suc A = (A ∪ {A})
3 uniun 3590 . . . . 5 (A ∪ {A}) = ( A {A})
42, 3eqtri 2057 . . . 4 suc A = ( A {A})
5 unisng 3588 . . . . 5 (A 𝑉 {A} = A)
65uneq2d 3091 . . . 4 (A 𝑉 → ( A {A}) = ( AA))
74, 6syl5eq 2081 . . 3 (A 𝑉 suc A = ( AA))
87eqeq1d 2045 . 2 (A 𝑉 → ( suc A = A ↔ ( AA) = A))
9 df-tr 3846 . . 3 (Tr A AA)
10 ssequn1 3107 . . 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 1242   wcel 1390  cun 2909  wss 2911  {csn 3367   cuni 3571  Tr wtr 3845  suc csuc 4068
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-suc 4074
This theorem is referenced by:  nlimsucg  4242
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