Theorem nlimsucg 4598

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 4426	. . . . . 6 |- ( Lim suc A -> Ord suc A )
2		ordsuc 4595	. . . . . 6 |- ( Ord A <-> Ord suc A )
3	1, 2	sylibr 134	. . . . 5 |- ( Lim suc A -> Ord A )
4		limuni 4427	. . . . 5 |- ( Lim suc A -> suc A = U. suc A )
5	3, 4	jca 306	. . . 4 |- ( Lim suc A -> ( Ord A /\ suc A = U. suc A ) )
6		ordtr 4409	. . . . . . . 8 |- ( Ord A -> Tr A )
7		unisucg 4445	. . . . . . . . 9 |- ( A e. V -> ( Tr A <-> U. suc A = A ) )
8	7	biimpa 296	. . . . . . . 8 |- ( ( A e. V /\ Tr A ) -> U. suc A = A )
9	6, 8	sylan2 286	. . . . . . 7 |- ( ( A e. V /\ Ord A ) -> U. suc A = A )
10	9	eqeq2d 2205	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A <-> suc A = A ) )
11		ordirr 4574	. . . . . . . . 9 |- ( Ord A -> -. A e. A )
12		eleq2 2257	. . . . . . . . . 10 |- ( suc A = A -> ( A e. suc A <-> A e. A ) )
13	12	notbid 668	. . . . . . . . 9 |- ( suc A = A -> ( -. A e. suc A <-> -. A e. A ) )
14	11, 13	syl5ibrcom 157	. . . . . . . 8 |- ( Ord A -> ( suc A = A -> -. A e. suc A ) )
15		sucidg 4447	. . . . . . . . 9 |- ( A e. V -> A e. suc A )
16	15	con3i 633	. . . . . . . 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 277	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = A -> -. A e. V ) )
19	10, 18	sylbid 150	. . . . 5 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A -> -. A e. V ) )
20	19	expimpd 363	. . . 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 625	. 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 104   = wceq 1364   e. wcel 2164   U.cuni 3835   Tr wtr 4127   Ord word 4393   Lim wlim 4395   suc csuc 4396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-ilim 4400  df-suc 4402

This theorem is referenced by: (None)