Theorem ordsuc 4547

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 4484	. 2 |- ( Ord A -> Ord suc A )
2		en2lp 4538	. . . . . . . . . 10 |- -. ( x e. A /\ A e. x )
3		eleq1 2233	. . . . . . . . . . . . 13 |- ( y = A -> ( y e. x <-> A e. x ) )
4	3	biimpac 296	. . . . . . . . . . . 12 |- ( ( y e. x /\ y = A ) -> A e. x )
5	4	anim2i 340	. . . . . . . . . . 11 |- ( ( x e. A /\ ( y e. x /\ y = A ) ) -> ( x e. A /\ A e. x ) )
6	5	expr 373	. . . . . . . . . 10 |- ( ( x e. A /\ y e. x ) -> ( y = A -> ( x e. A /\ A e. x ) ) )
7	2, 6	mtoi 659	. . . . . . . . 9 |- ( ( x e. A /\ y e. x ) -> -. y = A )
8	7	adantl 275	. . . . . . . 8 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> -. y = A )
9		elelsuc 4394	. . . . . . . . . . . . . . 15 |- ( x e. A -> x e. suc A )
10	9	adantr 274	. . . . . . . . . . . . . 14 |- ( ( x e. A /\ y e. x ) -> x e. suc A )
11		ordelss 4364	. . . . . . . . . . . . . 14 |- ( ( Ord suc A /\ x e. suc A ) -> x C_ suc A )
12	10, 11	sylan2 284	. . . . . . . . . . . . 13 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> x C_ suc A )
13	12	sseld 3146	. . . . . . . . . . . 12 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> ( y e. x -> y e. suc A ) )
14	13	expr 373	. . . . . . . . . . 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 377	. . . . . . . . 9 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. suc A )
17		elsuci 4388	. . . . . . . . 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 1344	. . . . . . 7 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. A )
20	19	ancom2s 561	. . . . . 6 |- ( ( Ord suc A /\ ( y e. x /\ x e. A ) ) -> y e. A )
21	20	ex 114	. . . . 5 |- ( Ord suc A -> ( ( y e. x /\ x e. A ) -> y e. A ) )
22	21	alrimivv 1868	. . . 4 |- ( Ord suc A -> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
23		dftr2 4089	. . . 4 |- ( Tr A <-> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
24	22, 23	sylibr 133	. . 3 |- ( Ord suc A -> Tr A )
25		sssucid 4400	. . . 4 |- A C_ suc A
26		trssord 4365	. . . 4 |- ( ( Tr A /\ A C_ suc A /\ Ord suc A ) -> Ord A )
27	25, 26	mp3an2 1320	. . 3 |- ( ( Tr A /\ Ord suc A ) -> Ord A )
28	24, 27	mpancom 420	. 2 |- ( Ord suc A -> Ord A )
29	1, 28	impbii 125	1 |- ( Ord A <-> Ord suc A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   A.wal 1346   = wceq 1348   e. wcel 2141   C_ wss 3121   Tr wtr 4087   Ord word 4347   suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-suc 4356

This theorem is referenced by:  nlimsucg  4550  ordpwsucss  4551