Theorem nnsucelsuc 6546

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4541, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4563. (Contributed by Jim Kingdon, 25-Aug-2019.)

Assertion

Ref	Expression
nnsucelsuc	|- ( B e. om -> ( A e. B <-> suc A e. suc B ) )

Proof of Theorem nnsucelsuc

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

Step	Hyp	Ref	Expression
1		eleq2 2257	. . . 4 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
2		suceq 4434	. . . . 5 |- ( x = (/) -> suc x = suc (/) )
3	2	eleq2d 2263	. . . 4 |- ( x = (/) -> ( suc A e. suc x <-> suc A e. suc (/) ) )
4	1, 3	imbi12d 234	. . 3 |- ( x = (/) -> ( ( A e. x -> suc A e. suc x ) <-> ( A e. (/) -> suc A e. suc (/) ) ) )
5		eleq2 2257	. . . 4 |- ( x = y -> ( A e. x <-> A e. y ) )
6		suceq 4434	. . . . 5 |- ( x = y -> suc x = suc y )
7	6	eleq2d 2263	. . . 4 |- ( x = y -> ( suc A e. suc x <-> suc A e. suc y ) )
8	5, 7	imbi12d 234	. . 3 |- ( x = y -> ( ( A e. x -> suc A e. suc x ) <-> ( A e. y -> suc A e. suc y ) ) )
9		eleq2 2257	. . . 4 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
10		suceq 4434	. . . . 5 |- ( x = suc y -> suc x = suc suc y )
11	10	eleq2d 2263	. . . 4 |- ( x = suc y -> ( suc A e. suc x <-> suc A e. suc suc y ) )
12	9, 11	imbi12d 234	. . 3 |- ( x = suc y -> ( ( A e. x -> suc A e. suc x ) <-> ( A e. suc y -> suc A e. suc suc y ) ) )
13		eleq2 2257	. . . 4 |- ( x = B -> ( A e. x <-> A e. B ) )
14		suceq 4434	. . . . 5 |- ( x = B -> suc x = suc B )
15	14	eleq2d 2263	. . . 4 |- ( x = B -> ( suc A e. suc x <-> suc A e. suc B ) )
16	13, 15	imbi12d 234	. . 3 |- ( x = B -> ( ( A e. x -> suc A e. suc x ) <-> ( A e. B -> suc A e. suc B ) ) )
17		noel 3451	. . . 4 |- -. A e. (/)
18	17	pm2.21i 647	. . 3 |- ( A e. (/) -> suc A e. suc (/) )
19		elsuci 4435	. . . . . . . 8 |- ( A e. suc y -> ( A e. y \/ A = y ) )
20	19	adantl 277	. . . . . . 7 |- ( ( ( A e. y -> suc A e. suc y ) /\ A e. suc y ) -> ( A e. y \/ A = y ) )
21		simpl 109	. . . . . . . 8 |- ( ( ( A e. y -> suc A e. suc y ) /\ A e. suc y ) -> ( A e. y -> suc A e. suc y ) )
22		suceq 4434	. . . . . . . . 9 |- ( A = y -> suc A = suc y )
23	22	a1i 9	. . . . . . . 8 |- ( ( ( A e. y -> suc A e. suc y ) /\ A e. suc y ) -> ( A = y -> suc A = suc y ) )
24	21, 23	orim12d 787	. . . . . . 7 |- ( ( ( A e. y -> suc A e. suc y ) /\ A e. suc y ) -> ( ( A e. y \/ A = y ) -> ( suc A e. suc y \/ suc A = suc y ) ) )
25	20, 24	mpd 13	. . . . . 6 |- ( ( ( A e. y -> suc A e. suc y ) /\ A e. suc y ) -> ( suc A e. suc y \/ suc A = suc y ) )
26		vex 2763	. . . . . . . 8 |- y e. _V
27	26	sucex 4532	. . . . . . 7 |- suc y e. _V
28	27	elsuc2 4439	. . . . . 6 |- ( suc A e. suc suc y <-> ( suc A e. suc y \/ suc A = suc y ) )
29	25, 28	sylibr 134	. . . . 5 |- ( ( ( A e. y -> suc A e. suc y ) /\ A e. suc y ) -> suc A e. suc suc y )
30	29	ex 115	. . . 4 |- ( ( A e. y -> suc A e. suc y ) -> ( A e. suc y -> suc A e. suc suc y ) )
31	30	a1i 9	. . 3 |- ( y e. om -> ( ( A e. y -> suc A e. suc y ) -> ( A e. suc y -> suc A e. suc suc y ) ) )
32	4, 8, 12, 16, 18, 31	finds 4633	. 2 |- ( B e. om -> ( A e. B -> suc A e. suc B ) )
33		nnon 4643	. . 3 |- ( B e. om -> B e. On )
34		onsucelsucr 4541	. . 3 |- ( B e. On -> ( suc A e. suc B -> A e. B ) )
35	33, 34	syl 14	. 2 |- ( B e. om -> ( suc A e. suc B -> A e. B ) )
36	32, 35	impbid 129	1 |- ( B e. om -> ( A e. B <-> suc A e. suc B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   e. wcel 2164   (/)c0 3447   Oncon0 4395   suc csuc 4397   omcom 4623

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-int 3872  df-tr 4129  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624

This theorem is referenced by:  nnsucsssuc  6547  nntri3or  6548  nnsucuniel  6550  nnaordi  6563  ennnfonelemhom  12575