Theorem ordsuc 4287

Description: The successor of an ordinal class is ordinal. (Contributed by NM, 3-Apr-1995.) (Constructive proof by Mario Carneiro and Jim Kingdon, 20-Jul-2019.)

Assertion

Ref	Expression
ordsuc	|- ( Ord A <-> Ord suc A )

Proof of Theorem ordsuc

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ordsucim 4226	. 2 |- ( Ord A -> Ord suc A )
2		en2lp 4278	. . . . . . . . . 10 |- -. ( x e. A /\ A e. x )
3		eleq1 2100	. . . . . . . . . . . . 13 |- ( y = A -> ( y e. x <-> A e. x ) )
4	3	biimpac 282	. . . . . . . . . . . 12 |- ( ( y e. x /\ y = A ) -> A e. x )
5	4	anim2i 324	. . . . . . . . . . 11 |- ( ( x e. A /\ ( y e. x /\ y = A ) ) -> ( x e. A /\ A e. x ) )
6	5	expr 357	. . . . . . . . . 10 |- ( ( x e. A /\ y e. x ) -> ( y = A -> ( x e. A /\ A e. x ) ) )
7	2, 6	mtoi 590	. . . . . . . . 9 |- ( ( x e. A /\ y e. x ) -> -. y = A )
8	7	adantl 262	. . . . . . . 8 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> -. y = A )
9		elelsuc 4146	. . . . . . . . . . . . . . 15 |- ( x e. A -> x e. suc A )
10	9	adantr 261	. . . . . . . . . . . . . 14 |- ( ( x e. A /\ y e. x ) -> x e. suc A )
11		ordelss 4116	. . . . . . . . . . . . . 14 |- ( ( Ord suc A /\ x e. suc A ) -> x C_ suc A )
12	10, 11	sylan2 270	. . . . . . . . . . . . 13 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> x C_ suc A )
13	12	sseld 2944	. . . . . . . . . . . 12 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> ( y e. x -> y e. suc A ) )
14	13	expr 357	. . . . . . . . . . 11 |- ( ( Ord suc A /\ x e. A ) -> ( y e. x -> ( y e. x -> y e. suc A ) ) )
15	14	pm2.43d 44	. . . . . . . . . 10 |- ( ( Ord suc A /\ x e. A ) -> ( y e. x -> y e. suc A ) )
16	15	impr 361	. . . . . . . . 9 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. suc A )
17		elsuci 4140	. . . . . . . . 9 |- ( y e. suc A -> ( y e. A \/ y = A ) )
18	16, 17	syl 14	. . . . . . . 8 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> ( y e. A \/ y = A ) )
19	8, 18	ecased 1239	. . . . . . 7 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. A )
20	19	ancom2s 500	. . . . . 6 |- ( ( Ord suc A /\ ( y e. x /\ x e. A ) ) -> y e. A )
21	20	ex 108	. . . . 5 |- ( Ord suc A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
22	21	alrimivv 1755	. . . 4 |- ( Ord suc A -> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
23		dftr2 3856	. . . 4 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
24	22, 23	sylibr 137	. . 3 |- ( Ord suc A -> Tr A )
25		sssucid 4152	. . . 4 |- A C_ suc A
26		trssord 4117	. . . 4 |- ( ( Tr A /\ A C_ suc A /\ Ord suc A ) -> Ord A )
27	25, 26	mp3an2 1220	. . 3 |- ( ( Tr A /\ Ord suc A ) -> Ord A )
28	24, 27	mpancom 399	. 2 |- ( Ord suc A -> Ord A )
29	1, 28	impbii 117	1 |- ( Ord A <-> Ord suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   A.wal 1241   = wceq 1243   e. wcel 1393   C_ wss 2917   Tr wtr 3854   Ord word 4099   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108

This theorem is referenced by:  nlimsucg  4290  ordpwsucss  4291