Theorem onunisuci 3783

Description: An ordinal number is equal to the union of its successor.

Hypothesis

Ref	Expression
on.1	|- A e. On

Assertion

Ref	Expression
onunisuci	|- U.suc A = A

Proof of Theorem onunisuci

Step	Hyp	Ref	Expression
1		on.1	. . 3 |- A e. On
2	1	ontrci 3775	. 2 |- Tr A
3	1	elisseti 2301	. . 3 |- A e. _V
4	3	unisuc 3741	. 2 |- (Tr A <-> U.suc A = A)
5	2, 4	mpbi 206	1 |- U.suc A = A

Syntax hints:   = wceq 1298   e. wcel 1300  U.cuni 3177  Tr wtr 3411  Oncon0 3657  suc csuc 3659

This theorem is referenced by:  tz7.44-2 5137  rankuni 5809

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-un 2600  df-in 2603  df-ss 2605  df-sn 3049  df-pr 3050  df-uni 3178  df-tr 3412  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663

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