Theorem unizlim 5844

Description: An ordinal equal to its own union is either zero or a limit ordinal. (Contributed by NM, 1-Oct-2003.)

Assertion

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

Proof of Theorem unizlim

Step	Hyp	Ref	Expression
1		df-ne 2795	. . . . . . 7 |- ( A =/= (/) <-> -. A = (/) )
2		df-lim 5728	. . . . . . . . 9 |- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
3	2	biimpri 218	. . . . . . . 8 |- ( ( Ord A /\ A =/= (/) /\ A = U. A ) -> Lim A )
4	3	3exp 1264	. . . . . . 7 |- ( Ord A -> ( A =/= (/) -> ( A = U. A -> Lim A ) ) )
5	1, 4	syl5bir 233	. . . . . 6 |- ( Ord A -> ( -. A = (/) -> ( A = U. A -> Lim A ) ) )
6	5	com23 86	. . . . 5 |- ( Ord A -> ( A = U. A -> ( -. A = (/) -> Lim A ) ) )
7	6	imp 445	. . . 4 |- ( ( Ord A /\ A = U. A ) -> ( -. A = (/) -> Lim A ) )
8	7	orrd 393	. . 3 |- ( ( Ord A /\ A = U. A ) -> ( A = (/) \/ Lim A ) )
9	8	ex 450	. 2 |- ( Ord A -> ( A = U. A -> ( A = (/) \/ Lim A ) ) )
10		uni0 4465	. . . . 5 |- U. (/) = (/)
11	10	eqcomi 2631	. . . 4 |- (/) = U. (/)
12		id 22	. . . 4 |- ( A = (/) -> A = (/) )
13		unieq 4444	. . . 4 |- ( A = (/) -> U. A = U. (/) )
14	11, 12, 13	3eqtr4a 2682	. . 3 |- ( A = (/) -> A = U. A )
15		limuni 5785	. . 3 |- ( Lim A -> A = U. A )
16	14, 15	jaoi 394	. 2 |- ( ( A = (/) \/ Lim A ) -> A = U. A )
17	9, 16	impbid1 215	1 |- ( Ord A -> ( A = U. A <-> ( A = (/) \/ Lim A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   =/= wne 2794   (/)c0 3915   U.cuni 4436   Ord word 5722   Lim wlim 5724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-lim 5728

This theorem is referenced by:  ordzsl  7045  oeeulem  7681  cantnfp1lem2  8576  cantnflem1  8586  cnfcom2lem  8598  ordcmp  32446