MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssonuni Structured version   Visualization version   GIF version

Theorem ssonuni 7799
Description: The union of a set of ordinal numbers is an ordinal number. Theorem 9 of [Suppes] p. 132. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 1-Nov-2003.)
Assertion
Ref Expression
ssonuni (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))

Proof of Theorem ssonuni
StepHypRef Expression
1 ssorduni 7798 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 uniexg 7759 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 elong 6394 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴))
51, 4imbitrrid 246 1 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2106  Vcvv 3478  wss 3963   cuni 4912  Ord word 6385  Oncon0 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390
This theorem is referenced by:  ssonunii  7800  onuni  7808  iunon  8378  onfununi  8380  oemapvali  9722  cardprclem  10017  carduni  10019  dfac12lem2  10183  ontgval  36414  onsupcl2  43214  onuniintrab  43215  onsupuni  43218  onsupcl3  43222  cantnfub2  43312
  Copyright terms: Public domain W3C validator