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Theorem ssorduni 7768
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 4912 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
2 ssel 3975 . . . . . . . . 9 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
3 onelss 6406 . . . . . . . . 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 4936 . . . . . . 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 5271 . . 3 (Tr 𝐴 ↔ ∀𝑥 𝐴𝑥 𝐴)
1311, 12sylibr 233 . 2 (𝐴 ⊆ On → Tr 𝐴)
14 onelon 6389 . . . . . . 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 3988 . 2 (𝐴 ⊆ On → 𝐴 ⊆ On)
20 ordon 7766 . . 3 Ord On
21 trssord 6381 . . . 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 3948   cuni 4908  Tr wtr 5265  Ord word 6363  Oncon0 6364
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 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  ssonuni  7769  ssonprc  7777  orduni  7779  onsucuni  7818  limuni3  7843  onfununi  8343  tfrlem8  8386  cofon1  8673  cofon2  8674  naddcllem  8677  onssnum  10037  unialeph  10098  cfslbn  10264  hsmexlem1  10423  inaprc  10833  bdayimaon  27203  noetasuplem4  27246  noetainflem4  27250  noeta2  27293  etasslt2  27323  scutbdaybnd2lim  27326  onsupneqmaxlim0  42055  onsupnmax  42059  onsupsucismax  42111  onsucunifi  42202
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