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

Theorem ssonuni 7485
 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 7484 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 uniexg 7450 . . 3 (𝐴𝑉 𝐴 ∈ V)
3 elong 6171 . . 3 ( 𝐴 ∈ V → ( 𝐴 ∈ On ↔ Ord 𝐴))
42, 3syl 17 . 2 (𝐴𝑉 → ( 𝐴 ∈ On ↔ Ord 𝐴))
51, 4syl5ibr 249 1 (𝐴𝑉 → (𝐴 ⊆ On → 𝐴 ∈ On))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2112  Vcvv 3444   ⊆ wss 3884  ∪ cuni 4803  Ord word 6162  Oncon0 6163 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167 This theorem is referenced by:  ssonunii  7486  onuni  7492  iunon  7963  onfununi  7965  oemapvali  9135  cardprclem  9396  carduni  9398  dfac12lem2  9559  ontgval  33887
 Copyright terms: Public domain W3C validator