Theorem nlimsucg 4309

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 4150	. . . . . 6 |- ( Lim suc A -> Ord suc A )
2		ordsuc 4306	. . . . . 6 |- ( Ord A <-> Ord suc A )
3	1, 2	sylibr 132	. . . . 5 |- ( Lim suc A -> Ord A )
4		limuni 4151	. . . . 5 |- ( Lim suc A -> suc A = U. suc A )
5	3, 4	jca 300	. . . 4 |- ( Lim suc A -> ( Ord A /\ suc A = U. suc A ) )
6		ordtr 4133	. . . . . . . 8 |- ( Ord A -> Tr A )
7		unisucg 4169	. . . . . . . . 9 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )
8	7	biimpa 290	. . . . . . . 8 |- ( ( A e. V /\ Tr A ) -> U. suc A = A )
9	6, 8	sylan2 280	. . . . . . 7 |- ( ( A e. V /\ Ord A ) -> U. suc A = A )
10	9	eqeq2d 2092	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A <-> suc A = A ) )
11		ordirr 4285	. . . . . . . . 9 |- ( Ord A -> -. A e. A )
12		eleq2 2142	. . . . . . . . . 10 |- ( suc A = A -> ( A e. suc A <-> A e. A ) )
13	12	notbid 624	. . . . . . . . 9 |- ( suc A = A -> ( -. A e. suc A <-> -. A e. A ) )
14	11, 13	syl5ibrcom 155	. . . . . . . 8 |- ( Ord A -> ( suc A = A -> -. A e. suc A ) )
15		sucidg 4171	. . . . . . . . 9 |- ( A e. V -> A e. suc A )
16	15	con3i 594	. . . . . . . 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 271	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = A -> -. A e. V ) )
19	10, 18	sylbid 148	. . . . 5 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A -> -. A e. V ) )
20	19	expimpd 355	. . . 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 586	. 2 |- ( A e. V -> ( A e. V -> -. Lim suc A ) )
23	22	pm2.43i 48	1 |- ( A e. V -> -. Lim suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   U.cuni 3601   Tr wtr 3875   Ord word 4117   Lim wlim 4119   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-ilim 4124  df-suc 4126

This theorem is referenced by: (None)