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Theorem ssorduni 4213
 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

Proof of Theorem ssorduni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3584 . . . . 5
2 ssel 2939 . . . . . . . . 9
3 onelss 4124 . . . . . . . . 9
42, 3syl6 29 . . . . . . . 8
5 anc2r 311 . . . . . . . 8
64, 5syl 14 . . . . . . 7
7 ssuni 3602 . . . . . . 7
86, 7syl8 65 . . . . . 6
98rexlimdv 2432 . . . . 5
101, 9syl5bi 141 . . . 4
1110ralrimiv 2391 . . 3
12 dftr3 3858 . . 3
1311, 12sylibr 137 . 2
14 onelon 4121 . . . . . . 7
1514ex 108 . . . . . 6
162, 15syl6 29 . . . . 5
1716rexlimdv 2432 . . . 4
181, 17syl5bi 141 . . 3
1918ssrdv 2951 . 2
20 ordon 4212 . . 3
21 trssord 4117 . . . 4
22213exp 1103 . . 3
2320, 22mpii 39 . 2
2413, 19, 23sylc 56 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wcel 1393  wral 2306  wrex 2307   wss 2917  cuni 3580   wtr 3854   word 4099  con0 4100 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  ssonuni  4214  orduni  4221  tfrlem8  5934  tfrexlem  5948
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