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Theorem unisucg 4339
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 4296 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
21unieqi 3749 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
3 uniun 3758 . . . . 5 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
42, 3eqtri 2160 . . . 4 suc 𝐴 = ( 𝐴 {𝐴})
5 unisng 3756 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
65uneq2d 3230 . . . 4 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
74, 6syl5eq 2184 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
87eqeq1d 2148 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
9 df-tr 4030 . . 3 (Tr 𝐴 𝐴𝐴)
10 ssequn1 3246 . . 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 1331  wcel 1480  cun 3069  wss 3071  {csn 3527   cuni 3739  Tr wtr 4029  suc csuc 4290
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-uni 3740  df-tr 4030  df-suc 4296
This theorem is referenced by:  onsucuni2  4482  nlimsucg  4484  ctmlemr  6996  nnsf  13347  peano4nninf  13348
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