Theorem unizlim 3786

Description: An ordinal equal to its own union is either zero or a limit ordinal.

Assertion

Ref	Expression
unizlim	|- (Ord A -> (A = U.A <-> (A = (/) \/ Lim A)))

Proof of Theorem unizlim

Step	Hyp	Ref	Expression
1		df-lim 3662	. . . . . . . . 9 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
2	1	biimpri 169	. . . . . . . 8 |- ((Ord A /\ A =/= (/) /\ A = U.A) -> Lim A)
3	2	3exp 1066	. . . . . . 7 |- (Ord A -> (A =/= (/) -> (A = U.A -> Lim A)))
4		df-ne 2019	. . . . . . 7 |- (A =/= (/) <-> -. A = (/))
5	3, 4	syl5ibr 224	. . . . . 6 |- (Ord A -> (-. A = (/) -> (A = U.A -> Lim A)))
6	5	com23 36	. . . . 5 |- (Ord A -> (A = U.A -> (-. A = (/) -> Lim A)))
7	6	imp 377	. . . 4 |- ((Ord A /\ A = U.A) -> (-. A = (/) -> Lim A))
8	7	orrd 250	. . 3 |- ((Ord A /\ A = U.A) -> (A = (/) \/ Lim A))
9	8	ex 402	. 2 |- (Ord A -> (A = U.A -> (A = (/) \/ Lim A)))
10		uni0 3205	. . . . 5 |- U.(/) = (/)
11	10	eqcomi 1888	. . . 4 |- (/) = U.(/)
12		id 73	. . . 4 |- (A = (/) -> A = (/))
13		unieq 3185	. . . 4 |- (A = (/) -> U.A = U.(/))
14	11, 12, 13	3eqtr4a 1954	. . 3 |- (A = (/) -> A = U.A)
15		limuni 3724	. . 3 |- (Lim A -> A = U.A)
16	14, 15	jaoi 368	. 2 |- ((A = (/) \/ Lim A) -> A = U.A)
17	9, 16	impbid1 575	1 |- (Ord A -> (A = U.A <-> (A = (/) \/ Lim A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   =/= wne 2017  (/)c0 2875  U.cuni 3177  Ord word 3656  Lim wlim 3658

This theorem is referenced by:  ordzsl 3927

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-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-sn 3049  df-uni 3178  df-lim 3662

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