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Theorem unisuc 4264
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 NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisuc.1 𝐴 ∈ V
Assertion
Ref Expression
unisuc (Tr 𝐴 suc 𝐴 = 𝐴)

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 3185 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 3959 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 4222 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 3685 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 3694 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 3691 . . . . 5 {𝐴} = 𝐴
87uneq2i 3166 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2119 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2102 . 2 ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
111, 2, 103bitr4i 211 1 (Tr 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1296  wcel 1445  Vcvv 2633  cun 3011  wss 3013  {csn 3466   cuni 3675  Tr wtr 3958  suc csuc 4216
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 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-pr 3473  df-uni 3676  df-tr 3959  df-suc 4222
This theorem is referenced by:  onunisuci  4283  ordsucunielexmid  4375  tfrexlem  6137  nnsucuniel  6296
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