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Theorem unisucg 4151
 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 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))

Proof of Theorem unisucg
StepHypRef Expression
1 df-suc 4108 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
21unieqi 3590 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
3 uniun 3599 . . . . 5 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
42, 3eqtri 2060 . . . 4 suc 𝐴 = ( 𝐴 {𝐴})
5 unisng 3597 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
65uneq2d 3097 . . . 4 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
74, 6syl5eq 2084 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
87eqeq1d 2048 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
9 df-tr 3855 . . 3 (Tr 𝐴 𝐴𝐴)
10 ssequn1 3113 . . 3 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
119, 10bitri 173 . 2 (Tr 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
128, 11syl6rbbr 188 1 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  {csn 3375  ∪ cuni 3580  Tr wtr 3854  suc 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:  onsucuni2  4288  nlimsucg  4290
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