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Theorem onsucuni 4618
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 4576 . 2  |-  ( A 
C_  On  ->  Ord  U. A )
2 ssid 3198 . . 3  |-  U. A  C_ 
U. A
3 ordunisssuc 4494 . . 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 3153   U.cuni 3828   Ord word 4390   Oncon0 4391   suc csuc 4393
This theorem is referenced by:  ordsucuni  4619  tz9.12lem3  7457  onssnum  7663  dfac12lem2  7766  ackbij1lem16  7857  cfslb2n  7890  hsmexlem1  8048
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-suc 4397
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