Theorem nlimsucg 7042

Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
nlimsucg	|- ( A e. V -> -. Lim suc A )

Proof of Theorem nlimsucg

Step	Hyp	Ref	Expression
1		limord 5784	. . . 4 |- ( Lim suc A -> Ord suc A )
2		ordsuc 7014	. . . 4 |- ( Ord A <-> Ord suc A )
3	1, 2	sylibr 224	. . 3 |- ( Lim suc A -> Ord A )
4		limuni 5785	. . 3 |- ( Lim suc A -> suc A = U. suc A )
5		ordunisuc 7032	. . . . 5 |- ( Ord A -> U. suc A = A )
6	5	eqeq2d 2632	. . . 4 |- ( Ord A -> ( suc A = U. suc A <-> suc A = A ) )
7		ordirr 5741	. . . . . 6 |- ( Ord A -> -. A e. A )
8		eleq2 2690	. . . . . . 7 |- ( suc A = A -> ( A e. suc A <-> A e. A ) )
9	8	notbid 308	. . . . . 6 |- ( suc A = A -> ( -. A e. suc A <-> -. A e. A ) )
10	7, 9	syl5ibrcom 237	. . . . 5 |- ( Ord A -> ( suc A = A -> -. A e. suc A ) )
11		sucidg 5803	. . . . . 6 |- ( A e. V -> A e. suc A )
12	11	con3i 150	. . . . 5 |- ( -. A e. suc A -> -. A e. V )
13	10, 12	syl6 35	. . . 4 |- ( Ord A -> ( suc A = A -> -. A e. V ) )
14	6, 13	sylbid 230	. . 3 |- ( Ord A -> ( suc A = U. suc A -> -. A e. V ) )
15	3, 4, 14	sylc 65	. 2 |- ( Lim suc A -> -. A e. V )
16	15	con2i 134	1 |- ( A e. V -> -. Lim suc A )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   e. wcel 1990   U.cuni 4436   Ord word 5722   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-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:  tz7.44-2  7503  rankxpsuc  8745  dfrdg2  31701  dfrdg4  32058