ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ordunisuc2r Structured version   GIF version

Theorem ordunisuc2r 4162
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 2536 . . . . . . . . 9 x V
21sucid 4076 . . . . . . . 8 x suc x
3 elunii 3537 . . . . . . . 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 1323 . . . . 5 (x(x A → suc x A) → x(x Ax A))
7 df-ral 2287 . . . . 5 (x A suc x Ax(x A → suc x A))
8 dfss2 2912 . . . . 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 4085 . . 3 (Ord A AA)
1210, 11jctird 300 . 2 (Ord A → (x A suc x A → (A A AA)))
13 eqss 2938 . 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 1314   = wceq 1373   wcel 1375  wral 2282  wss 2895   cuni 3532  Ord word 4023  suc csuc 4026
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-sn 3333  df-uni 3533  df-tr 3807  df-iord 4027  df-suc 4031
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator