Theorem dflim3 7047

Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
dflim3	|- ( Lim A <-> ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) )

Distinct variable group:   x, A

Proof of Theorem dflim3

Step	Hyp	Ref	Expression
1		df-lim 5728	. 2 |- ( Lim A <-> ( Ord A /\ A =/= (/) /\ A = U. A ) )
2		3anass 1042	. 2 |- ( ( Ord A /\ A =/= (/) /\ A = U. A ) <-> ( Ord A /\ ( A =/= (/) /\ A = U. A ) ) )
3		df-ne 2795	. . . . . 6 |- ( A =/= (/) <-> -. A = (/) )
4	3	a1i 11	. . . . 5 |- ( Ord A -> ( A =/= (/) <-> -. A = (/) ) )
5		orduninsuc 7043	. . . . 5 |- ( Ord A -> ( A = U. A <-> -. E. x e. On A = suc x ) )
6	4, 5	anbi12d 747	. . . 4 |- ( Ord A -> ( ( A =/= (/) /\ A = U. A ) <-> ( -. A = (/) /\ -. E. x e. On A = suc x ) ) )
7		ioran 511	. . . 4 |- ( -. ( A = (/) \/ E. x e. On A = suc x ) <-> ( -. A = (/) /\ -. E. x e. On A = suc x ) )
8	6, 7	syl6bbr 278	. . 3 |- ( Ord A -> ( ( A =/= (/) /\ A = U. A ) <-> -. ( A = (/) \/ E. x e. On A = suc x ) ) )
9	8	pm5.32i 669	. 2 |- ( ( Ord A /\ ( A =/= (/) /\ A = U. A ) ) <-> ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) )
10	1, 2, 9	3bitri 286	1 |- ( Lim A <-> ( Ord A /\ -. ( A = (/) \/ E. x e. On A = suc x ) ) )

Syntax hints:   -. wn 3   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   =/= wne 2794   E.wrex 2913   (/)c0 3915   U.cuni 4436   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729

This theorem is referenced by:  nlimon  7051  tfinds  7059  oalimcl  7640  omlimcl  7658  r1wunlim  9559