Theorem nlimsucg 3107

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 3060	. . . . . . 7 |- (Ord A <-> Ord suc A)
2		ordirr 2961	. . . . . . 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 1531	. . . . . . 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 3047	. . . . . . . . 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 1545	. . . . . . 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 3084	. . . . . 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 3024	. . . 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 787	. . . 4 |- ((Ord suc A /\ suc A =/= (/) /\ suc A = U.suc A) -> Ord suc A)
19		df-lim 2948	. . . 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 774   = wceq 954   e. wcel 956   =/= wne 1582  (/)c0 2276  U.cuni 2498  Ord word 2942  Lim wlim 2944  suc csuc 2945

This theorem is referenced by:  tz7.44-2 3920  rankxpsuc 4695

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949

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