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Theorem ssorduni 7766
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 4913 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
2 ssel 3976 . . . . . . . . 9 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
3 onelss 6407 . . . . . . . . 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 4937 . . . . . . 7 ((𝑥𝑦𝑦𝐴) → 𝑥 𝐴)
86, 7syl8 76 . . . . . 6 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 𝐴)))
98rexlimdv 3154 . . . . 5 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 𝐴))
101, 9biimtrid 241 . . . 4 (𝐴 ⊆ On → (𝑥 𝐴𝑥 𝐴))
1110ralrimiv 3146 . . 3 (𝐴 ⊆ On → ∀𝑥 𝐴𝑥 𝐴)
12 dftr3 5272 . . 3 (Tr 𝐴 ↔ ∀𝑥 𝐴𝑥 𝐴)
1311, 12sylibr 233 . 2 (𝐴 ⊆ On → Tr 𝐴)
14 onelon 6390 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
1514ex 414 . . . . . 6 (𝑦 ∈ On → (𝑥𝑦𝑥 ∈ On))
162, 15syl6 35 . . . . 5 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 ∈ On)))
1716rexlimdv 3154 . . . 4 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 ∈ On))
181, 17biimtrid 241 . . 3 (𝐴 ⊆ On → (𝑥 𝐴𝑥 ∈ On))
1918ssrdv 3989 . 2 (𝐴 ⊆ On → 𝐴 ⊆ On)
20 ordon 7764 . . 3 Ord On
21 trssord 6382 . . . 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 3062  wrex 3071  wss 3949   cuni 4909  Tr wtr 5266  Ord word 6364  Oncon0 6365
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369
This theorem is referenced by:  ssonuni  7767  ssonprc  7775  orduni  7777  onsucuni  7816  limuni3  7841  onfununi  8341  tfrlem8  8384  cofon1  8671  cofon2  8672  naddcllem  8675  onssnum  10035  unialeph  10096  cfslbn  10262  hsmexlem1  10421  inaprc  10831  bdayimaon  27196  noetasuplem4  27239  noetainflem4  27243  noeta2  27286  etasslt2  27315  scutbdaybnd2lim  27318  onsupneqmaxlim0  41973  onsupnmax  41977  onsupsucismax  42029  onsucunifi  42120
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