Theorem ordsuc 4306

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 4244	. 2 |- ( Ord A -> Ord suc A )
2		en2lp 4297	. . . . . . . . . 10 |- -. ( x e. A /\ A e. x )
3		eleq1 2141	. . . . . . . . . . . . 13 |- ( y = A -> ( y e. x <-> A e. x ) )
4	3	biimpac 292	. . . . . . . . . . . 12 |- ( ( y e. x /\ y = A ) -> A e. x )
5	4	anim2i 334	. . . . . . . . . . 11 |- ( ( x e. A /\ ( y e. x /\ y = A ) ) -> ( x e. A /\ A e. x ) )
6	5	expr 367	. . . . . . . . . 10 |- ( ( x e. A /\ y e. x ) -> ( y = A -> ( x e. A /\ A e. x ) ) )
7	2, 6	mtoi 622	. . . . . . . . 9 |- ( ( x e. A /\ y e. x ) -> -. y = A )
8	7	adantl 271	. . . . . . . 8 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> -. y = A )
9		elelsuc 4164	. . . . . . . . . . . . . . 15 |- ( x e. A -> x e. suc A )
10	9	adantr 270	. . . . . . . . . . . . . 14 |- ( ( x e. A /\ y e. x ) -> x e. suc A )
11		ordelss 4134	. . . . . . . . . . . . . 14 |- ( ( Ord suc A /\ x e. suc A ) -> x C_ suc A )
12	10, 11	sylan2 280	. . . . . . . . . . . . 13 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> x C_ suc A )
13	12	sseld 2998	. . . . . . . . . . . 12 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> ( y e. x -> y e. suc A ) )
14	13	expr 367	. . . . . . . . . . 11 |- ( ( Ord suc A /\ x e. A ) -> ( y e. x -> ( y e. x -> y e. suc A ) ) )
15	14	pm2.43d 49	. . . . . . . . . 10 |- ( ( Ord suc A /\ x e. A ) -> ( y e. x -> y e. suc A ) )
16	15	impr 371	. . . . . . . . 9 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. suc A )
17		elsuci 4158	. . . . . . . . 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 1280	. . . . . . 7 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. A )
20	19	ancom2s 530	. . . . . 6 |- ( ( Ord suc A /\ ( y e. x /\ x e. A ) ) -> y e. A )
21	20	ex 113	. . . . 5 |- ( Ord suc A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
22	21	alrimivv 1796	. . . 4 |- ( Ord suc A -> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
23		dftr2 3877	. . . 4 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
24	22, 23	sylibr 132	. . 3 |- ( Ord suc A -> Tr A )
25		sssucid 4170	. . . 4 |- A C_ suc A
26		trssord 4135	. . . 4 |- ( ( Tr A /\ A C_ suc A /\ Ord suc A ) -> Ord A )
27	25, 26	mp3an2 1256	. . 3 |- ( ( Tr A /\ Ord suc A ) -> Ord A )
28	24, 27	mpancom 413	. 2 |- ( Ord suc A -> Ord A )
29	1, 28	impbii 124	1 |- ( Ord A <-> Ord suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   C_ wss 2973   Tr wtr 3875   Ord word 4117   suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-suc 4126

This theorem is referenced by:  nlimsucg  4309  ordpwsucss  4310