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

Theorem ssorduni 7762
Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. Lemma 2.7 of [Schloeder] p. 4. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
ssorduni (𝐴 ⊆ On → Ord 𝐴)

Proof of Theorem ssorduni
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 4911 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
2 ssel 3974 . . . . . . . . 9 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
3 onelss 6403 . . . . . . . . 9 (𝑦 ∈ On → (𝑥𝑦𝑥𝑦))
42, 3syl6 35 . . . . . . . 8 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥𝑦)))
5 anc2r 555 . . . . . . . 8 ((𝑦𝐴 → (𝑥𝑦𝑥𝑦)) → (𝑦𝐴 → (𝑥𝑦 → (𝑥𝑦𝑦𝐴))))
64, 5syl 17 . . . . . . 7 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦 → (𝑥𝑦𝑦𝐴))))
7 ssuni 4935 . . . . . . 7 ((𝑥𝑦𝑦𝐴) → 𝑥 𝐴)
86, 7syl8 76 . . . . . 6 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 𝐴)))
98rexlimdv 3153 . . . . 5 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 𝐴))
101, 9biimtrid 241 . . . 4 (𝐴 ⊆ On → (𝑥 𝐴𝑥 𝐴))
1110ralrimiv 3145 . . 3 (𝐴 ⊆ On → ∀𝑥 𝐴𝑥 𝐴)
12 dftr3 5270 . . 3 (Tr 𝐴 ↔ ∀𝑥 𝐴𝑥 𝐴)
1311, 12sylibr 233 . 2 (𝐴 ⊆ On → Tr 𝐴)
14 onelon 6386 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
1514ex 413 . . . . . 6 (𝑦 ∈ On → (𝑥𝑦𝑥 ∈ On))
162, 15syl6 35 . . . . 5 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 ∈ On)))
1716rexlimdv 3153 . . . 4 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 ∈ On))
181, 17biimtrid 241 . . 3 (𝐴 ⊆ On → (𝑥 𝐴𝑥 ∈ On))
1918ssrdv 3987 . 2 (𝐴 ⊆ On → 𝐴 ⊆ On)
20 ordon 7760 . . 3 Ord On
21 trssord 6378 . . . 4 ((Tr 𝐴 𝐴 ⊆ On ∧ Ord On) → Ord 𝐴)
22213exp 1119 . . 3 (Tr 𝐴 → ( 𝐴 ⊆ On → (Ord On → Ord 𝐴)))
2320, 22mpii 46 . 2 (Tr 𝐴 → ( 𝐴 ⊆ On → Ord 𝐴))
2413, 19, 23sylc 65 1 (𝐴 ⊆ On → Ord 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wral 3061  wrex 3070  wss 3947   cuni 4907  Tr wtr 5264  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
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:  ssonuni  7763  ssonprc  7771  orduni  7773  onsucuni  7812  limuni3  7837  onfununi  8337  tfrlem8  8380  cofon1  8667  cofon2  8668  naddcllem  8671  onssnum  10031  unialeph  10092  cfslbn  10258  hsmexlem1  10417  inaprc  10827  bdayimaon  27185  noetasuplem4  27228  noetainflem4  27232  noeta2  27275  etasslt2  27304  scutbdaybnd2lim  27307  onsupneqmaxlim0  41958  onsupnmax  41962  onsupsucismax  42014  onsucunifi  42105
  Copyright terms: Public domain W3C validator