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Theorem unisucg 4410
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-tr 4099 . . 3 (Tr 𝐴 𝐴𝐴)
2 ssequn1 3305 . . 3 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
31, 2bitri 184 . 2 (Tr 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
4 df-suc 4367 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
54unieqi 3817 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
6 uniun 3826 . . . . 5 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
75, 6eqtri 2198 . . . 4 suc 𝐴 = ( 𝐴 {𝐴})
8 unisng 3824 . . . . 5 (𝐴𝑉 {𝐴} = 𝐴)
98uneq2d 3289 . . . 4 (𝐴𝑉 → ( 𝐴 {𝐴}) = ( 𝐴𝐴))
107, 9eqtrid 2222 . . 3 (𝐴𝑉 suc 𝐴 = ( 𝐴𝐴))
1110eqeq1d 2186 . 2 (𝐴𝑉 → ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴))
123, 11bitr4id 199 1 (𝐴𝑉 → (Tr 𝐴 suc 𝐴 = 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wcel 2148  cun 3127  wss 3129  {csn 3591   cuni 3807  Tr wtr 4098  suc csuc 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-uni 3808  df-tr 4099  df-suc 4367
This theorem is referenced by:  onsucuni2  4559  nlimsucg  4561  ctmlemr  7100  nnnninfeq2  7120  nnsf  14377  peano4nninf  14378
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