Theorem unizlim 2364

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 2204	. . . . . . . 8 |- (Lim A <-> (Ord A /\ -. A = (/) /\ A = U.A))
2	1	biimpr 134	. . . . . . 7 |- ((Ord A /\ -. A = (/) /\ A = U.A) -> Lim A)
3	2	3exp 611	. . . . . 6 |- (Ord A -> (-. A = (/) -> (A = U.A -> Lim A)))
4	3	com23 32	. . . . 5 |- (Ord A -> (A = U.A -> (-. A = (/) -> Lim A)))
5	4	imp 277	. . . 4 |- ((Ord A /\ A = U.A) -> (-. A = (/) -> Lim A))
6	5	orrd 203	. . 3 |- ((Ord A /\ A = U.A) -> (A = (/) \/ Lim A))
7	6	exp 291	. 2 |- (Ord A -> (A = U.A -> (A = (/) \/ Lim A)))
8		uni0 1938	. . . . . 6 |- U.(/) = (/)
9	8	cleqcomi 1105	. . . . 5 |- (/) = U.(/)
10		id 9	. . . . . 6 |- (A = (/) -> A = (/))
11		unieq 1927	. . . . . 6 |- (A = (/) -> U.A = U.(/))
12	10, 11	cleq12d 1115	. . . . 5 |- (A = (/) -> (A = U.A <-> (/) = U.(/)))
13	9, 12	mpbiri 169	. . . 4 |- (A = (/) -> A = U.A)
14		limuni 2284	. . . 4 |- (Lim A -> A = U.A)
15	13, 14	jaoi 275	. . 3 |- ((A = (/) \/ Lim A) -> A = U.A)
16	15	a1i 7	. 2 |- (Ord A -> ((A = (/) \/ Lim A) -> A = U.A))
17	7, 16	impbid 397	1 |- (Ord A -> (A = U.A <-> (A = (/) \/ Lim A)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196   /\ w3a 581   = wceq 1091  (/)c0 1707  U.cuni 1919  Ord word 2198  Lim wlim 2200

This theorem is referenced by:  ordzsl 2366

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-sn 1811  df-pr 1812  df-uni 1920  df-lim 2204

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