Theorem ordsuc 4599

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 4536	. 2 |- ( Ord A -> Ord suc A )
2		en2lp 4590	. . . . . . . . . 10 |- -. ( x e. A /\ A e. x )
3		eleq1 2259	. . . . . . . . . . . . 13 |- ( y = A -> ( y e. x <-> A e. x ) )
4	3	biimpac 298	. . . . . . . . . . . 12 |- ( ( y e. x /\ y = A ) -> A e. x )
5	4	anim2i 342	. . . . . . . . . . 11 |- ( ( x e. A /\ ( y e. x /\ y = A ) ) -> ( x e. A /\ A e. x ) )
6	5	expr 375	. . . . . . . . . 10 |- ( ( x e. A /\ y e. x ) -> ( y = A -> ( x e. A /\ A e. x ) ) )
7	2, 6	mtoi 665	. . . . . . . . 9 |- ( ( x e. A /\ y e. x ) -> -. y = A )
8	7	adantl 277	. . . . . . . 8 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> -. y = A )
9		elelsuc 4444	. . . . . . . . . . . . . . 15 |- ( x e. A -> x e. suc A )
10	9	adantr 276	. . . . . . . . . . . . . 14 |- ( ( x e. A /\ y e. x ) -> x e. suc A )
11		ordelss 4414	. . . . . . . . . . . . . 14 |- ( ( Ord suc A /\ x e. suc A ) -> x C_ suc A )
12	10, 11	sylan2 286	. . . . . . . . . . . . 13 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> x C_ suc A )
13	12	sseld 3182	. . . . . . . . . . . 12 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> ( y e. x -> y e. suc A ) )
14	13	expr 375	. . . . . . . . . . 11 |- ( ( Ord suc A /\ x e. A ) -> ( y e. x -> ( y e. x -> y e. suc A ) ) )
15	14	pm2.43d 50	. . . . . . . . . 10 |- ( ( Ord suc A /\ x e. A ) -> ( y e. x -> y e. suc A ) )
16	15	impr 379	. . . . . . . . 9 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. suc A )
17		elsuci 4438	. . . . . . . . 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 1360	. . . . . . 7 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. A )
20	19	ancom2s 566	. . . . . 6 |- ( ( Ord suc A /\ ( y e. x /\ x e. A ) ) -> y e. A )
21	20	ex 115	. . . . 5 |- ( Ord suc A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
22	21	alrimivv 1889	. . . 4 |- ( Ord suc A -> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
23		dftr2 4133	. . . 4 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
24	22, 23	sylibr 134	. . 3 |- ( Ord suc A -> Tr A )
25		sssucid 4450	. . . 4 |- A C_ suc A
26		trssord 4415	. . . 4 |- ( ( Tr A /\ A C_ suc A /\ Ord suc A ) -> Ord A )
27	25, 26	mp3an2 1336	. . 3 |- ( ( Tr A /\ Ord suc A ) -> Ord A )
28	24, 27	mpancom 422	. 2 |- ( Ord suc A -> Ord A )
29	1, 28	impbii 126	1 |- ( Ord A <-> Ord suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   e. wcel 2167   C_ wss 3157   Tr wtr 4131   Ord word 4397   suc csuc 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-suc 4406

This theorem is referenced by:  nlimsucg  4602  ordpwsucss  4603