Theorem nlimsucg 4419

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 4255	. . . . . 6 |- ( Lim suc A -> Ord suc A )
2		ordsuc 4416	. . . . . 6 |- ( Ord A <-> Ord suc A )
3	1, 2	sylibr 133	. . . . 5 |- ( Lim suc A -> Ord A )
4		limuni 4256	. . . . 5 |- ( Lim suc A -> suc A = U. suc A )
5	3, 4	jca 302	. . . 4 |- ( Lim suc A -> ( Ord A /\ suc A = U. suc A ) )
6		ordtr 4238	. . . . . . . 8 |- ( Ord A -> Tr A )
7		unisucg 4274	. . . . . . . . 9 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )
8	7	biimpa 292	. . . . . . . 8 |- ( ( A e. V /\ Tr A ) -> U. suc A = A )
9	6, 8	sylan2 282	. . . . . . 7 |- ( ( A e. V /\ Ord A ) -> U. suc A = A )
10	9	eqeq2d 2111	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A <-> suc A = A ) )
11		ordirr 4395	. . . . . . . . 9 |- ( Ord A -> -. A e. A )
12		eleq2 2163	. . . . . . . . . 10 |- ( suc A = A -> ( A e. suc A <-> A e. A ) )
13	12	notbid 633	. . . . . . . . 9 |- ( suc A = A -> ( -. A e. suc A <-> -. A e. A ) )
14	11, 13	syl5ibrcom 156	. . . . . . . 8 |- ( Ord A -> ( suc A = A -> -. A e. suc A ) )
15		sucidg 4276	. . . . . . . . 9 |- ( A e. V -> A e. suc A )
16	15	con3i 602	. . . . . . . 8 |- ( -. A e. suc A -> -. A e. V )
17	14, 16	syl6 33	. . . . . . 7 |- ( Ord A -> ( suc A = A -> -. A e. V ) )
18	17	adantl 273	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = A -> -. A e. V ) )
19	10, 18	sylbid 149	. . . . 5 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A -> -. A e. V ) )
20	19	expimpd 358	. . . 4 |- ( A e. V -> ( ( Ord A /\ suc A = U. suc A ) -> -. A e. V ) )
21	5, 20	syl5 32	. . 3 |- ( A e. V -> ( Lim suc A -> -. A e. V ) )
22	21	con2d 594	. 2 |- ( A e. V -> ( A e. V -> -. Lim suc A ) )
23	22	pm2.43i 49	1 |- ( A e. V -> -. Lim suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   U.cuni 3683   Tr wtr 3966   Ord word 4222   Lim wlim 4224   suc csuc 4225

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-uni 3684  df-tr 3967  df-iord 4226  df-ilim 4229  df-suc 4231

This theorem is referenced by: (None)