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Theorem ordunisuc2r 4187
Description: An ordinal which contains the successor of each of its members is equal to its union. (Contributed by Jim Kingdon, 14-Nov-2018.)
Assertion
Ref Expression
ordunisuc2r (Ord A → (x A suc x AA = A))
Distinct variable group:   x,A

Proof of Theorem ordunisuc2r
StepHypRef Expression
1 vex 2537 . . . . . . . . 9 x V
21sucid 4101 . . . . . . . 8 x suc x
3 elunii 3558 . . . . . . . 8 ((x suc x suc x A) → x A)
42, 3mpan 402 . . . . . . 7 (suc x Ax A)
54imim2i 12 . . . . . 6 ((x A → suc x A) → (x Ax A))
65alimi 1324 . . . . 5 (x(x A → suc x A) → x(x Ax A))
7 df-ral 2288 . . . . 5 (x A suc x Ax(x A → suc x A))
8 dfss2 2910 . . . . 5 (A Ax(x Ax A))
96, 7, 83imtr4i 190 . . . 4 (x A suc x AA A)
109a1i 9 . . 3 (Ord A → (x A suc x AA A))
11 orduniss 4110 . . 3 (Ord A AA)
1210, 11jctird 300 . 2 (Ord A → (x A suc x A → (A A AA)))
13 eqss 2936 . 2 (A = A ↔ (A A AA))
1412, 13syl6ibr 151 1 (Ord A → (x A suc x AA = A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228   wcel 1375  wral 2283  wss 2893   cuni 3553  Ord word 4046  suc csuc 4049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-sn 3355  df-uni 3554  df-tr 3828  df-iord 4050  df-suc 4055
This theorem is referenced by: (None)
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