Theorem ordsuc 4540

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 4477	. 2 |- ( Ord A -> Ord suc A )
2		en2lp 4531	. . . . . . . . . 10 |- -. ( x e. A /\ A e. x )
3		eleq1 2229	. . . . . . . . . . . . 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 654	. . . . . . . . 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 4387	. . . . . . . . . . . . . . 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 4357	. . . . . . . . . . . . . 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 3141	. . . . . . . . . . . 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 4381	. . . . . . . . 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 1339	. . . . . . 7 |- ( ( Ord suc A /\ ( x e. A /\ y e. x ) ) -> y e. A )
20	19	ancom2s 556	. . . . . 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 1863	. . . 4 |- ( Ord suc A -> A. y A. x ( ( y e. x /\ x e. A ) -> y e. A ) )
23		dftr2 4082	. . . 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 4393	. . . 4 |- A C_ suc A
26		trssord 4358	. . . 4 |- ( ( Tr A /\ A C_ suc A /\ Ord suc A ) -> Ord A )
27	25, 26	mp3an2 1315	. . 3 |- ( ( Tr A /\ Ord suc A ) -> Ord A )
28	24, 27	mpancom 419	. 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 698   A.wal 1341   = wceq 1343   e. wcel 2136   C_ wss 3116   Tr wtr 4080   Ord word 4340   suc csuc 4343

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-suc 4349

This theorem is referenced by:  nlimsucg  4543  ordpwsucss  4544