Theorem nlimsucg 4693

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 4521	. . . . . 6 |- ( Lim suc A -> Ord suc A )
2		ordsuc 4690	. . . . . 6 |- ( Ord A <-> Ord suc A )
3	1, 2	sylibr 134	. . . . 5 |- ( Lim suc A -> Ord A )
4		limuni 4522	. . . . 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 4504	. . . . . . . 8 |- ( Ord A -> Tr A )
7		unisucg 4540	. . . . . . . . 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 2246	. . . . . 6 |- ( ( A e. V /\ Ord A ) -> ( suc A = U. suc A <-> suc A = A ) )
11		ordirr 4669	. . . . . . . . 9 |- ( Ord A -> -. A e. A )
12		eleq2 2298	. . . . . . . . . 10 |- ( suc A = A -> ( A e. suc A <-> A e. A ) )
13	12	notbid 673	. . . . . . . . 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 4542	. . . . . . . . 9 |- ( A e. V -> A e. suc A )
16	15	con3i 637	. . . . . . . 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 629	. 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 1398   e. wcel 2205   U.cuni 3919   Tr wtr 4213   Ord word 4488   Lim wlim 4490   suc csuc 4491

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-sn 3700  df-pr 3701  df-uni 3920  df-tr 4214  df-iord 4492  df-ilim 4495  df-suc 4497

This theorem is referenced by: (None)