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Theorem onsucuni 4619
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 4577 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 ssid 3197 . . 3  |-  U. A  C_ 
U. A
3 ordunisssuc 4495 . . 3  |-  ( ( A  C_  On  /\  Ord  U. A )  ->  ( U. A  C_  U. A  <->  A 
C_  suc  U. A ) )
42, 3mpbii 202 . 2  |-  ( ( A  C_  On  /\  Ord  U. A )  ->  A  C_ 
suc  U. A )
51, 4mpdan 649 1  |-  ( A 
C_  On  ->  A  C_  suc  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3152   U.cuni 3827   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  ordsucuni  4620  tz9.12lem3  7461  onssnum  7667  dfac12lem2  7770  ackbij1lem16  7861  cfslb2n  7894  hsmexlem1  8052
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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