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Theorem ssorduni 7488
Description: The union of a class of ordinal numbers is ordinal. Proposition 7.19 of [TakeutiZaring] p. 40. (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 4841 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦𝐴 𝑥𝑦)
2 ssel 3965 . . . . . . . . 9 (𝐴 ⊆ On → (𝑦𝐴𝑦 ∈ On))
3 onelss 6231 . . . . . . . . 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 4859 . . . . . . 7 ((𝑥𝑦𝑦𝐴) → 𝑥 𝐴)
86, 7syl8 76 . . . . . 6 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 𝐴)))
98rexlimdv 3288 . . . . 5 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 𝐴))
101, 9syl5bi 243 . . . 4 (𝐴 ⊆ On → (𝑥 𝐴𝑥 𝐴))
1110ralrimiv 3186 . . 3 (𝐴 ⊆ On → ∀𝑥 𝐴𝑥 𝐴)
12 dftr3 5173 . . 3 (Tr 𝐴 ↔ ∀𝑥 𝐴𝑥 𝐴)
1311, 12sylibr 235 . 2 (𝐴 ⊆ On → Tr 𝐴)
14 onelon 6214 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
1514ex 413 . . . . . 6 (𝑦 ∈ On → (𝑥𝑦𝑥 ∈ On))
162, 15syl6 35 . . . . 5 (𝐴 ⊆ On → (𝑦𝐴 → (𝑥𝑦𝑥 ∈ On)))
1716rexlimdv 3288 . . . 4 (𝐴 ⊆ On → (∃𝑦𝐴 𝑥𝑦𝑥 ∈ On))
181, 17syl5bi 243 . . 3 (𝐴 ⊆ On → (𝑥 𝐴𝑥 ∈ On))
1918ssrdv 3977 . 2 (𝐴 ⊆ On → 𝐴 ⊆ On)
20 ordon 7486 . . 3 Ord On
21 trssord 6206 . . . 4 ((Tr 𝐴 𝐴 ⊆ On ∧ Ord On) → Ord 𝐴)
22213exp 1113 . . 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 2107  wral 3143  wrex 3144  wss 3940   cuni 4837  Tr wtr 5169  Ord word 6188  Oncon0 6189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-tr 5170  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6192  df-on 6193
This theorem is referenced by:  ssonuni  7489  ssonprc  7495  orduni  7497  onsucuni  7531  limuni3  7555  onfununi  7969  tfrlem8  8011  onssnum  9455  unialeph  9516  cfslbn  9678  hsmexlem1  9837  inaprc  10247  bdayimaon  33081  nosupbday  33089  noetalem3  33103  noetalem4  33104
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