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Theorem unisucg 4204
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 4161 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
21unieqi 3637 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
3 uniun 3646 . . . . 5 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
42, 3eqtri 2103 . . . 4 suc 𝐴 = ( 𝐴 {𝐴})
5 unisng 3644 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
65uneq2d 3138 . . . 4 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
74, 6syl5eq 2127 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
87eqeq1d 2091 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
9 df-tr 3902 . . 3 (Tr 𝐴 𝐴𝐴)
10 ssequn1 3154 . . 3 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
119, 10bitri 182 . 2 (Tr 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
128, 11syl6rbbr 197 1 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wcel 1434  cun 2982  wss 2984  {csn 3422   cuni 3627  Tr wtr 3901  suc csuc 4155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-suc 4161
This theorem is referenced by:  onsucuni2  4342  nlimsucg  4344
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