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Theorem ssorduni 4163
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 (A ⊆ On → Ord A)

Proof of Theorem ssorduni
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3558 . . . . 5 (x Ay A x y)
2 ssel 2916 . . . . . . . . 9 (A ⊆ On → (y Ay On))
3 onelss 4073 . . . . . . . . 9 (y On → (x yxy))
42, 3syl6 29 . . . . . . . 8 (A ⊆ On → (y A → (x yxy)))
5 anc2r 311 . . . . . . . 8 ((y A → (x yxy)) → (y A → (x y → (xy y A))))
64, 5syl 14 . . . . . . 7 (A ⊆ On → (y A → (x y → (xy y A))))
7 ssuni 3576 . . . . . . 7 ((xy y A) → x A)
86, 7syl8 65 . . . . . 6 (A ⊆ On → (y A → (x yx A)))
98rexlimdv 2410 . . . . 5 (A ⊆ On → (y A x yx A))
101, 9syl5bi 141 . . . 4 (A ⊆ On → (x Ax A))
1110ralrimiv 2369 . . 3 (A ⊆ On → x Ax A)
12 dftr3 3832 . . 3 (Tr Ax Ax A)
1311, 12sylibr 137 . 2 (A ⊆ On → Tr A)
14 onelon 4070 . . . . . . 7 ((y On x y) → x On)
1514ex 108 . . . . . 6 (y On → (x yx On))
162, 15syl6 29 . . . . 5 (A ⊆ On → (y A → (x yx On)))
1716rexlimdv 2410 . . . 4 (A ⊆ On → (y A x yx On))
181, 17syl5bi 141 . . 3 (A ⊆ On → (x Ax On))
1918ssrdv 2928 . 2 (A ⊆ On → A ⊆ On)
20 ordon 4162 . . 3 Ord On
21 trssord 4066 . . . 4 ((Tr A A ⊆ On Ord On) → Ord A)
22213exp 1089 . . 3 (Tr A → ( A ⊆ On → (Ord On → Ord A)))
2320, 22mpii 39 . 2 (Tr A → ( A ⊆ On → Ord A))
2413, 19, 23sylc 56 1 (A ⊆ On → Ord A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1374  wral 2284  wrex 2285  wss 2894   cuni 3554  Tr wtr 3828  Ord word 4048  Oncon0 4049
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-3an 875  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-rex 2290  df-v 2537  df-in 2901  df-ss 2908  df-uni 3555  df-tr 3829  df-iord 4052  df-on 4054
This theorem is referenced by:  ssonuni  4164  orduni  4171  tfrlem8  5856  tfrexlem  5870
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