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

Theorem ssonuni 7763
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 7762 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 uniexg 7726 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 elong 6369 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴))
51, 4imbitrrid 245 1 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2106  Vcvv 3474  wss 3947   cuni 4907  Ord word 6360  Oncon0 6361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365
This theorem is referenced by:  ssonunii  7764  onuni  7772  iunon  8335  onfununi  8337  oemapvali  9675  cardprclem  9970  carduni  9972  dfac12lem2  10135  ontgval  35304  onsupcl2  41959  onuniintrab  41960  onsupuni  41963  onsupcl3  41967  cantnfub2  42057
  Copyright terms: Public domain W3C validator