Theorem nlimsucg 4290

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 4132	. . . . . 6 |- ( Lim suc A -> Ord suc A )
2		ordsuc 4287	. . . . . 6 |- ( Ord A <-> Ord suc A )
3	1, 2	sylibr 137	. . . . 5 |- ( Lim suc A -> Ord A )
4		limuni 4133	. . . . 5 |- ( Lim suc A -> suc A = U. suc A )
5	3, 4	jca 290	. . . 4 |- ( Lim suc A -> ( Ord A /\ suc A = U. suc A ) )
6		ordtr 4115	. . . . . . . 8 |- ( Ord A -> Tr A )
7		unisucg 4151	. . . . . . . . 9 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )
8	7	biimpa 280	. . . . . . . 8 |- ( ( A e. V /\ Tr A ) -> U. suc A = A )
9	6, 8	sylan2 270	. . . . . . 7 |- ( ( A e. V /\ Ord A ) -> U. suc A = A )
10	9	eqeq2d 2051	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A <-> suc A = A ) )
11		ordirr 4267	. . . . . . . . 9 |- ( Ord A -> -. A e. A )
12		eleq2 2101	. . . . . . . . . 10 |- ( suc A = A -> ( A e. suc A <-> A e. A ) )
13	12	notbid 592	. . . . . . . . 9 |- ( suc A = A -> ( -. A e. suc A <-> -. A e. A ) )
14	11, 13	syl5ibrcom 146	. . . . . . . 8 |- ( Ord A -> ( suc A = A -> -. A e. suc A ) )
15		sucidg 4153	. . . . . . . . 9 |- ( A e. V -> A e. suc A )
16	15	con3i 562	. . . . . . . 8 |- ( -. A e. suc A -> -. A e. V )
17	14, 16	syl6 29	. . . . . . 7 |- ( Ord A -> ( suc A = A -> -. A e. V ) )
18	17	adantl 262	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = A -> -. A e. V ) )
19	10, 18	sylbid 139	. . . . 5 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A -> -. A e. V ) )
20	19	expimpd 345	. . . 4 |- ( A e. V -> ( ( Ord A /\ suc A = U. suc A ) -> -. A e. V ) )
21	5, 20	syl5 28	. . 3 |- ( A e. V -> ( Lim suc A -> -. A e. V ) )
22	21	con2d 554	. 2 |- ( A e. V -> ( A e. V -> -. Lim suc A ) )
23	22	pm2.43i 43	1 |- ( A e. V -> -. Lim suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   U.cuni 3580   Tr wtr 3854   Ord word 4099   Lim wlim 4101   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-ilim 4106  df-suc 4108

This theorem is referenced by: (None)