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Theorem ssorduni 4179
Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ssorduni 
C_  On  Ord  U.

Proof of Theorem ssorduni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3575 . . . . 5  U.
2 ssel 2933 . . . . . . . . 9 
C_  On  On
3 onelss 4090 . . . . . . . . 9  On 
C_
42, 3syl6 29 . . . . . . . 8 
C_  On  C_
5 anc2r 311 . . . . . . . 8  C_  C_
64, 5syl 14 . . . . . . 7 
C_  On  C_
7 ssuni 3593 . . . . . . 7  C_  C_  U.
86, 7syl8 65 . . . . . 6 
C_  On  C_  U.
98rexlimdv 2426 . . . . 5 
C_  On 
C_  U.
101, 9syl5bi 141 . . . 4 
C_  On  U. 
C_  U.
1110ralrimiv 2385 . . 3 
C_  On  U.  C_  U.
12 dftr3 3849 . . 3  Tr 
U.  U.  C_  U.
1311, 12sylibr 137 . 2 
C_  On  Tr  U.
14 onelon 4087 . . . . . . 7  On  On
1514ex 108 . . . . . 6  On  On
162, 15syl6 29 . . . . 5 
C_  On  On
1716rexlimdv 2426 . . . 4 
C_  On  On
181, 17syl5bi 141 . . 3 
C_  On  U.  On
1918ssrdv 2945 . 2 
C_  On  U.  C_  On
20 ordon 4178 . . 3  Ord  On
21 trssord 4083 . . . 4  Tr  U.  U.  C_  On  Ord  On  Ord  U.
22213exp 1102 . . 3  Tr 
U.  U.  C_  On  Ord  On  Ord  U.
2320, 22mpii 39 . 2  Tr 
U.  U.  C_  On  Ord  U.
2413, 19, 23sylc 56 1 
C_  On  Ord  U.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wcel 1390  wral 2300  wrex 2301    C_ wss 2911   U.cuni 3571   Tr wtr 3845   Ord word 4065   Oncon0 4066
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-3an 886  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-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071
This theorem is referenced by:  ssonuni  4180  orduni  4187  tfrlem8  5875  tfrexlem  5889
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