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Theorem ssonuni 4164
 Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. (Contributed by NM, 1-Nov-2003.)
Assertion
Ref Expression
ssonuni (A 𝑉 → (A ⊆ On → A On))

Proof of Theorem ssonuni
StepHypRef Expression
1 ssorduni 4163 . 2 (A ⊆ On → Ord A)
2 uniexg 4125 . . 3 (A 𝑉 A V)
3 elong 4059 . . 3 ( A V → ( A On ↔ Ord A))
42, 3syl 14 . 2 (A 𝑉 → ( A On ↔ Ord A))
51, 4syl5ibr 145 1 (A 𝑉 → (A ⊆ On → A On))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1374  Vcvv 2535   ⊆ wss 2894  ∪ cuni 3554  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-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-un 4120 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:  ssonunii  4165  onun2  4166  onuni  4170  iunon  5821
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