Theorem onsucuni 3075

Description: A class of ordinal numbers is a subclass of the successor of its union. Similar to Proposition 7.26 of [TakeutiZaring] p. 41.

Assertion

Ref	Expression
onsucuni	|- (A (_ On -> A (_ suc U.A)

Proof of Theorem onsucuni

Step	Hyp	Ref	Expression
1		ssorduni 2983	. 2 |- (A (_ On -> Ord U.A)
2		ssid 2070	. . 3 |- U.A (_ U.A
3		ordunisssuc 3073	. . 3 |- ((A (_ On /\ Ord U.A) -> (U.A (_ U.A <-> A (_ suc U.A))
4	2, 3	mpbii 193	. 2 |- ((A (_ On /\ Ord U.A) -> A (_ suc U.A)
5	1, 4	mpdan 702	1 |- (A (_ On -> A (_ suc U.A)

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2037  U.cuni 2493  Ord word 2937  Oncon0 2938  suc csuc 2940

This theorem is referenced by:  ordsucuni 3076  tz9.12lem3 4633

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944

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