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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 7799 . 2 (𝐴 ⊆ On → Ord 𝐴)
2 ssid 4006 . . 3 𝐴 𝐴
3 ordunisssuc 6490 . . 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 3951   cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  ordsucuni  7849  cofon1  8710  cofon2  8711  naddcllem  8714  tz9.12lem3  9829  onssnum  10080  dfac12lem2  10185  ackbij1lem16  10274  cfslb2n  10308  hsmexlem1  10466  noeta2  27829  etasslt2  27859  zs12bday  28424  cantnfub2  43335  onsucunifi  43383
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