Theorem limsssuc 7050

Description: A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)

Assertion

Ref	Expression
limsssuc	|- ( Lim A -> ( A C_ B <-> A C_ suc B ) )

Proof of Theorem limsssuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssucid 5802	. . 3 |- B C_ suc B
2		sstr2 3610	. . 3 |- ( A C_ B -> ( B C_ suc B -> A C_ suc B ) )
3	1, 2	mpi 20	. 2 |- ( A C_ B -> A C_ suc B )
4		eleq1 2689	. . . . . . . . . . . 12 |- ( x = B -> ( x e. A <-> B e. A ) )
5	4	biimpcd 239	. . . . . . . . . . 11 |- ( x e. A -> ( x = B -> B e. A ) )
6		limsuc 7049	. . . . . . . . . . . . . 14 |- ( Lim A -> ( B e. A <-> suc B e. A ) )
7	6	biimpa 501	. . . . . . . . . . . . 13 |- ( ( Lim A /\ B e. A ) -> suc B e. A )
8		limord 5784	. . . . . . . . . . . . . . . 16 |- ( Lim A -> Ord A )
9	8	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( Lim A /\ B e. A ) -> Ord A )
10		ordelord 5745	. . . . . . . . . . . . . . . . 17 |- ( ( Ord A /\ B e. A ) -> Ord B )
11	8, 10	sylan 488	. . . . . . . . . . . . . . . 16 |- ( ( Lim A /\ B e. A ) -> Ord B )
12		ordsuc 7014	. . . . . . . . . . . . . . . 16 |- ( Ord B <-> Ord suc B )
13	11, 12	sylib 208	. . . . . . . . . . . . . . 15 |- ( ( Lim A /\ B e. A ) -> Ord suc B )
14		ordtri1 5756	. . . . . . . . . . . . . . 15 |- ( ( Ord A /\ Ord suc B ) -> ( A C_ suc B <-> -. suc B e. A ) )
15	9, 13, 14	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( Lim A /\ B e. A ) -> ( A C_ suc B <-> -. suc B e. A ) )
16	15	con2bid 344	. . . . . . . . . . . . 13 |- ( ( Lim A /\ B e. A ) -> ( suc B e. A <-> -. A C_ suc B ) )
17	7, 16	mpbid 222	. . . . . . . . . . . 12 |- ( ( Lim A /\ B e. A ) -> -. A C_ suc B )
18	17	ex 450	. . . . . . . . . . 11 |- ( Lim A -> ( B e. A -> -. A C_ suc B ) )
19	5, 18	sylan9r 690	. . . . . . . . . 10 |- ( ( Lim A /\ x e. A ) -> ( x = B -> -. A C_ suc B ) )
20	19	con2d 129	. . . . . . . . 9 |- ( ( Lim A /\ x e. A ) -> ( A C_ suc B -> -. x = B ) )
21	20	ex 450	. . . . . . . 8 |- ( Lim A -> ( x e. A -> ( A C_ suc B -> -. x = B ) ) )
22	21	com23 86	. . . . . . 7 |- ( Lim A -> ( A C_ suc B -> ( x e. A -> -. x = B ) ) )
23	22	imp31 448	. . . . . 6 |- ( ( ( Lim A /\ A C_ suc B ) /\ x e. A ) -> -. x = B )
24		ssel2 3598	. . . . . . . . . 10 |- ( ( A C_ suc B /\ x e. A ) -> x e. suc B )
25		vex 3203	. . . . . . . . . . 11 |- x e. _V
26	25	elsuc 5794	. . . . . . . . . 10 |- ( x e. suc B <-> ( x e. B \/ x = B ) )
27	24, 26	sylib 208	. . . . . . . . 9 |- ( ( A C_ suc B /\ x e. A ) -> ( x e. B \/ x = B ) )
28	27	ord 392	. . . . . . . 8 |- ( ( A C_ suc B /\ x e. A ) -> ( -. x e. B -> x = B ) )
29	28	con1d 139	. . . . . . 7 |- ( ( A C_ suc B /\ x e. A ) -> ( -. x = B -> x e. B ) )
30	29	adantll 750	. . . . . 6 |- ( ( ( Lim A /\ A C_ suc B ) /\ x e. A ) -> ( -. x = B -> x e. B ) )
31	23, 30	mpd 15	. . . . 5 |- ( ( ( Lim A /\ A C_ suc B ) /\ x e. A ) -> x e. B )
32	31	ex 450	. . . 4 |- ( ( Lim A /\ A C_ suc B ) -> ( x e. A -> x e. B ) )
33	32	ssrdv 3609	. . 3 |- ( ( Lim A /\ A C_ suc B ) -> A C_ B )
34	33	ex 450	. 2 |- ( Lim A -> ( A C_ suc B -> A C_ B ) )
35	3, 34	impbid2 216	1 |- ( Lim A -> ( A C_ B <-> A C_ suc B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   Ord word 5722   Lim wlim 5724   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729

This theorem is referenced by:  cardlim  8798