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Theorem onsucuni 4748
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 
C_  On  ->  A  C_  suc  U. A )

Proof of Theorem onsucuni
StepHypRef Expression
1 ssorduni 4706 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 ssid 3310 . . 3  |-  U. A  C_ 
U. A
3 ordunisssuc 4624 . . 3  |-  ( ( A  C_  On  /\  Ord  U. A )  ->  ( U. A  C_  U. A  <->  A 
C_  suc  U. A ) )
42, 3mpbii 203 . 2  |-  ( ( A  C_  On  /\  Ord  U. A )  ->  A  C_ 
suc  U. A )
51, 4mpdan 650 1  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    C_ wss 3263   U.cuni 3957   Ord word 4521   Oncon0 4522   suc csuc 4524
This theorem is referenced by:  ordsucuni  4749  tz9.12lem3  7648  onssnum  7854  dfac12lem2  7957  ackbij1lem16  8048  cfslb2n  8081  hsmexlem1  8239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-tr 4244  df-eprel 4435  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-suc 4528
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