Theorem unizlim 3958

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-ne 2297	. . . . . . 7 |- (A =/= (/) <-> -. A = (/))
2		df-lim 3848	. . . . . . . . 9 |- (Lim A <-> (Ord A /\ A =/= (/) /\ A = U.A))
3	2	biimpri 243	. . . . . . . 8 |- ((Ord A /\ A =/= (/) /\ A = U.A) -> Lim A)
4	3	3exp 1344	. . . . . . 7 |- (Ord A -> (A =/= (/) -> (A = U.A -> Lim A)))
5	1, 4	syl5bir 272	. . . . . 6 |- (Ord A -> (-. A = (/) -> (A = U.A -> Lim A)))
6	5	com23 68	. . . . 5 |- (Ord A -> (A = U.A -> (-. A = (/) -> Lim A)))
7	6	imp 489	. . . 4 |- ((Ord A /\ A = U.A) -> (-. A = (/) -> Lim A))
8	7	orrd 442	. . 3 |- ((Ord A /\ A = U.A) -> (A = (/) \/ Lim A))
9	8	ex 494	. 2 |- (Ord A -> (A = U.A -> (A = (/) \/ Lim A)))
10		uni0 3423	. . . . 5 |- U.(/) = (/)
11	10	eqcomi 2174	. . . 4 |- (/) = U.(/)
12		id 15	. . . 4 |- (A = (/) -> A = (/))
13		unieq 3407	. . . 4 |- (A = (/) -> U.A = U.(/))
14	11, 12, 13	3eqtr4a 2231	. . 3 |- (A = (/) -> A = U.A)
15		limuni 3902	. . 3 |- (Lim A -> A = U.A)
16	14, 15	jaoi 549	. 2 |- ((A = (/) \/ Lim A) -> A = U.A)
17	9, 16	impbid1 251	1 |- (Ord A -> (A = U.A <-> (A = (/) \/ Lim A)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 231   \/ wo 432   /\ wa 433   /\ w3a 1130   = wceq 1615   =/= wne 2295  (/)c0 3114  U.cuni 3398  Ord word 3842  Lim wlim 3844

This theorem is referenced by:  ordzsl 4098

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1621  ax-gen 1622  ax-8 1623  ax-9 1624  ax-10 1625  ax-11 1626  ax-12 1627  ax-17 1634  ax-4 1637  ax-5o 1639  ax-6o 1642  ax-9o 1792  ax-10o 1810  ax-16 1883  ax-11o 1893  ax-ext 2152

This theorem depends on definitions:  df-bi 232  df-or 434  df-an 435  df-3an 1132  df-ex 1645  df-sb 1845  df-clab 2158  df-cleq 2163  df-clel 2166  df-ne 2297  df-ral 2389  df-rex 2390  df-v 2571  df-dif 2862  df-in 2866  df-ss 2868  df-nul 3115  df-sn 3274  df-uni 3399  df-lim 3848

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