Theorem nlimsucg 3102

Description: A successor is not a limit ordinal.

Assertion

Ref	Expression
nlimsucg	|- (A e. B -> -. Lim suc A)

Proof of Theorem nlimsucg

Step	Hyp	Ref	Expression
1		ordsuc 3055	. . . . . . 7 |- (Ord A <-> Ord suc A)
2		ordirr 2956	. . . . . . 7 |- (Ord suc A -> -. suc A e. suc A)
3	1, 2	sylbi 199	. . . . . 6 |- (Ord A -> -. suc A e. suc A)
4	3	adantl 388	. . . . 5 |- ((A e. B /\ Ord A) -> -. suc A e. suc A)
5		eleq1 1526	. . . . . . 7 |- (suc A = U.suc A -> (suc A e. suc A <-> U.suc A e. suc A))
6		pm3.27 323	. . . . . . . 8 |- ((A e. B /\ U.suc A = A) -> U.suc A = A)
7		sucidg 3042	. . . . . . . . 9 |- (A e. B -> A e. suc A)
8	7	adantr 389	. . . . . . . 8 |- ((A e. B /\ U.suc A = A) -> A e. suc A)
9	6, 8	eqeltrd 1540	. . . . . . 7 |- ((A e. B /\ U.suc A = A) -> U.suc A e. suc A)
10	5, 9	syl5cbir 211	. . . . . 6 |- ((A e. B /\ U.suc A = A) -> (suc A = U.suc A -> suc A e. suc A))
11		ordunisuc 3079	. . . . . 6 |- (Ord A -> U.suc A = A)
12	10, 11	sylan2 451	. . . . 5 |- ((A e. B /\ Ord A) -> (suc A = U.suc A -> suc A e. suc A))
13	4, 12	mtod 108	. . . 4 |- ((A e. B /\ Ord A) -> -. suc A = U.suc A)
14	13	ex 373	. . 3 |- (A e. B -> (Ord A -> -. suc A = U.suc A))
15		limuni 3019	. . . 4 |- (Lim suc A -> suc A = U.suc A)
16	15	a1i 8	. . 3 |- (A e. B -> (Lim suc A -> suc A = U.suc A))
17	14, 16	nsyld 117	. 2 |- (A e. B -> (Ord A -> -. Lim suc A))
18		3simp1 786	. . . 4 |- ((Ord suc A /\ suc A =/= (/) /\ suc A = U.suc A) -> Ord suc A)
19		df-lim 2943	. . . 4 |- (Lim suc A <-> (Ord suc A /\ suc A =/= (/) /\ suc A = U.suc A))
20	18, 19, 1	3imtr4 219	. . 3 |- (Lim suc A -> Ord A)
21	20	con3i 98	. 2 |- (-. Ord A -> -. Lim suc A)
22	17, 21	pm2.61d1 128	1 |- (A e. B -> -. Lim suc A)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  (/)c0 2270  U.cuni 2493  Ord word 2937  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  tz7.44-2 3914  rankxpsuc 4687

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944

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