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Theorem ordunisuc2r 4204
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 2554 . . . . . . . . 9 x V
21sucid 4119 . . . . . . . 8 x suc x
3 elunii 3575 . . . . . . . 8 ((x suc x suc x A) → x A)
42, 3mpan 400 . . . . . . 7 (suc x Ax A)
54imim2i 12 . . . . . 6 ((x A → suc x A) → (x Ax A))
65alimi 1341 . . . . 5 (x(x A → suc x A) → x(x Ax A))
7 df-ral 2305 . . . . 5 (x A suc x Ax(x A → suc x A))
8 dfss2 2928 . . . . 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 4127 . . 3 (Ord A AA)
1210, 11jctird 300 . 2 (Ord A → (x A suc x A → (A A AA)))
13 eqss 2954 . 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 1240   = wceq 1242   wcel 1390  wral 2300  wss 2911   cuni 3570  Ord word 4064  suc csuc 4067
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3372  df-uni 3571  df-tr 3845  df-iord 4068  df-suc 4073
This theorem is referenced by: (None)
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