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Theorem ssonuni 7028
 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 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))

Proof of Theorem ssonuni
StepHypRef Expression
1 ssorduni 7027 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 uniexg 6997 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 elong 5769 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴))
51, 4syl5ibr 236 1 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  ∪ cuni 4468  Ord word 5760  Oncon0 5761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765 This theorem is referenced by:  ssonunii  7029  onuni  7035  iunon  7481  onfununi  7483  oemapvali  8619  cardprclem  8843  carduni  8845  dfac12lem2  9004  ontgval  32555
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