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Theorem unisucg 4306
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 4263 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
21unieqi 3716 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
3 uniun 3725 . . . . 5 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
42, 3eqtri 2138 . . . 4 suc 𝐴 = ( 𝐴 {𝐴})
5 unisng 3723 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
65uneq2d 3200 . . . 4 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
74, 6syl5eq 2162 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
87eqeq1d 2126 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
9 df-tr 3997 . . 3 (Tr 𝐴 𝐴𝐴)
10 ssequn1 3216 . . 3 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
119, 10bitri 183 . 2 (Tr 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
128, 11syl6rbbr 198 1 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1316  wcel 1465  cun 3039  wss 3041  {csn 3497   cuni 3706  Tr wtr 3996  suc csuc 4257
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-uni 3707  df-tr 3997  df-suc 4263
This theorem is referenced by:  onsucuni2  4449  nlimsucg  4451  ctmlemr  6961  nnsf  13126  peano4nninf  13127
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