Theorem unizlim 3108

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 2948	. . . . . . . . 9 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
2	1	biimpr 152	. . . . . . . 8 |- ((Ord A /\ A =/= (/) /\ A = U.A) -> Lim A)
3	2	3exp 831	. . . . . . 7 |- (Ord A -> (A =/= (/) -> (A = U.A -> Lim A)))
4		df-ne 1584	. . . . . . 7 |- (A =/= (/) <-> -. A = (/))
5	3, 4	syl5ibr 207	. . . . . 6 |- (Ord A -> (-. A = (/) -> (A = U.A -> Lim A)))
6	5	com23 32	. . . . 5 |- (Ord A -> (A = U.A -> (-. A = (/) -> Lim A)))
7	6	imp 350	. . . 4 |- ((Ord A /\ A = U.A) -> (-. A = (/) -> Lim A))
8	7	orrd 233	. . 3 |- ((Ord A /\ A = U.A) -> (A = (/) \/ Lim A))
9	8	ex 373	. 2 |- (Ord A -> (A = U.A -> (A = (/) \/ Lim A)))
10		uni0 2520	. . . . 5 |- U.(/) = (/)
11	10	eqcomi 1476	. . . 4 |- (/) = U.(/)
12		id 59	. . . 4 |- (A = (/) -> A = (/))
13		unieq 2505	. . . 4 |- (A = (/) -> U.A = U.(/))
14	11, 12, 13	3eqtr4a 1529	. . 3 |- (A = (/) -> A = U.A)
15		limuni 3024	. . 3 |- (Lim A -> A = U.A)
16	14, 15	jaoi 341	. 2 |- ((A = (/) \/ Lim A) -> A = U.A)
17	9, 16	impbid1 516	1 |- (Ord A -> (A = U.A <-> (A = (/) \/ Lim A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 774   = wceq 954   =/= wne 1582  (/)c0 2276  U.cuni 2498  Ord word 2942  Lim wlim 2944

This theorem is referenced by:  ordzsl 3111

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-nul 2277  df-sn 2408  df-uni 2499  df-lim 2948

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