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Theorem ssorduni 7718
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 4874 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
2 ssel 3942 . . . . . . . . 9 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
3 onelss 6364 . . . . . . . . 9 (𝑦 ∈ On → (𝑥𝑦𝑥𝑦))
42, 3syl6 35 . . . . . . . 8 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥𝑦)))
5 anc2r 556 . . . . . . . 8 ((𝑦𝐴 → (𝑥𝑦𝑥𝑦)) → (𝑦𝐴 → (𝑥𝑦 → (𝑥𝑦𝑦𝐴))))
64, 5syl 17 . . . . . . 7 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦 → (𝑥𝑦𝑦𝐴))))
7 ssuni 4898 . . . . . . 7 ((𝑥𝑦𝑦𝐴) → 𝑥 𝐴)
86, 7syl8 76 . . . . . 6 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 𝐴)))
98rexlimdv 3151 . . . . 5 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 𝐴))
101, 9biimtrid 241 . . . 4 (𝐴 ⊆ On → (𝑥 𝐴𝑥 𝐴))
1110ralrimiv 3143 . . 3 (𝐴 ⊆ On → ∀𝑥 𝐴𝑥 𝐴)
12 dftr3 5233 . . 3 (Tr 𝐴 ↔ ∀𝑥 𝐴𝑥 𝐴)
1311, 12sylibr 233 . 2 (𝐴 ⊆ On → Tr 𝐴)
14 onelon 6347 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
1514ex 414 . . . . . 6 (𝑦 ∈ On → (𝑥𝑦𝑥 ∈ On))
162, 15syl6 35 . . . . 5 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 ∈ On)))
1716rexlimdv 3151 . . . 4 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 ∈ On))
181, 17biimtrid 241 . . 3 (𝐴 ⊆ On → (𝑥 𝐴𝑥 ∈ On))
1918ssrdv 3955 . 2 (𝐴 ⊆ On → 𝐴 ⊆ On)
20 ordon 7716 . . 3 Ord On
21 trssord 6339 . . . 4 ((Tr 𝐴 𝐴 ⊆ On ∧ Ord On) → Ord 𝐴)
22213exp 1120 . . 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 397  wcel 2107  wral 3065  wrex 3074  wss 3915   cuni 4870  Tr wtr 5227  Ord word 6321  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  ssonuni  7719  ssonprc  7727  orduni  7729  onsucuni  7768  limuni3  7793  onfununi  8292  tfrlem8  8335  cofon1  8623  cofon2  8624  naddcllem  8627  onssnum  9983  unialeph  10044  cfslbn  10210  hsmexlem1  10369  inaprc  10779  bdayimaon  27057  noetasuplem4  27100  noetainflem4  27104  noeta2  27146  etasslt2  27175  scutbdaybnd2lim  27178  onsupneqmaxlim0  41587  onsupnmax  41591  onsupsucismax  41643  onsucunifi  41715
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