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Theorem ordunisuc2r 4189
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  suc  U.
Distinct variable group:   ,

Proof of Theorem ordunisuc2r
StepHypRef Expression
1 vex 2538 . . . . . . . . 9 
_V
21sucid 4103 . . . . . . . 8 
suc
3 elunii 3559 . . . . . . . 8  suc  suc  U.
42, 3mpan 402 . . . . . . 7  suc  U.
54imim2i 12 . . . . . 6  suc  U.
65alimi 1324 . . . . 5  suc  U.
7 df-ral 2289 . . . . 5  suc  suc
8 dfss2 2911 . . . . 5 
C_  U.  U.
96, 7, 83imtr4i 190 . . . 4  suc 
C_  U.
109a1i 9 . . 3  Ord  suc  C_  U.
11 orduniss 4112 . . 3  Ord  U.  C_
1210, 11jctird 300 . 2  Ord  suc 
C_  U.  U.  C_
13 eqss 2937 . 2  U. 
C_  U.  U.  C_
1412, 13syl6ibr 151 1  Ord  suc  U.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1226   wceq 1228   wcel 1374  wral 2284    C_ wss 2894   U.cuni 3554   Ord word 4048   suc csuc 4051
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-uni 3555  df-tr 3829  df-iord 4052  df-suc 4057
This theorem is referenced by: (None)
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