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Theorem unisucg 4274
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 4231 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
21unieqi 3693 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
3 uniun 3702 . . . . 5 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
42, 3eqtri 2120 . . . 4 suc 𝐴 = ( 𝐴 {𝐴})
5 unisng 3700 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
65uneq2d 3177 . . . 4 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
74, 6syl5eq 2144 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
87eqeq1d 2108 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
9 df-tr 3967 . . 3 (Tr 𝐴 𝐴𝐴)
10 ssequn1 3193 . . 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 1299  wcel 1448  cun 3019  wss 3021  {csn 3474   cuni 3683  Tr wtr 3966  suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-suc 4231
This theorem is referenced by:  onsucuni2  4417  nlimsucg  4419  ctmlemr  6908  nnsf  12783  peano4nninf  12784
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