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Theorem onsucuni 7848
Description: A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41. (Contributed by NM, 19-Sep-2003.)
Assertion
Ref Expression
onsucuni (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)

Proof of Theorem onsucuni
StepHypRef Expression
1 ssorduni 7798 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 ssid 4018 . . 3 𝐴 𝐴
3 ordunisssuc 6492 . . 3 ((𝐴 ⊆ On ∧ Ord 𝐴) → ( 𝐴 𝐴𝐴 ⊆ suc 𝐴))
42, 3mpbii 233 . 2 ((𝐴 ⊆ On ∧ Ord 𝐴) → 𝐴 ⊆ suc 𝐴)
51, 4mpdan 687 1 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wss 3963   cuni 4912  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  ordsucuni  7849  cofon1  8709  cofon2  8710  naddcllem  8713  tz9.12lem3  9827  onssnum  10078  dfac12lem2  10183  ackbij1lem16  10272  cfslb2n  10306  hsmexlem1  10464  noeta2  27844  etasslt2  27874  zs12bday  28439  cantnfub2  43312  onsucunifi  43360
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