Theorem nnsucelsuc 6491

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4507, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4529. (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 2241	. . . 4 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
2		suceq 4402	. . . . 5 |- ( x = (/) -> suc x = suc (/) )
3	2	eleq2d 2247	. . . 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 2241	. . . 4 |- ( x = y -> ( A e. x <-> A e. y ) )
6		suceq 4402	. . . . 5 |- ( x = y -> suc x = suc y )
7	6	eleq2d 2247	. . . 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 2241	. . . 4 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
10		suceq 4402	. . . . 5 |- ( x = suc y -> suc x = suc suc y )
11	10	eleq2d 2247	. . . 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 2241	. . . 4 |- ( x = B -> ( A e. x <-> A e. B ) )
14		suceq 4402	. . . . 5 |- ( x = B -> suc x = suc B )
15	14	eleq2d 2247	. . . 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 3426	. . . 4 |- -. A e. (/)
18	17	pm2.21i 646	. . 3 |- ( A e. (/) -> suc A e. suc (/) )
19		elsuci 4403	. . . . . . . 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 4402	. . . . . . . . 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 786	. . . . . . 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 2740	. . . . . . . 8 |- y e. _V
27	26	sucex 4498	. . . . . . 7 |- suc y e. _V
28	27	elsuc2 4407	. . . . . 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 4599	. 2 |- ( B e. om -> ( A e. B -> suc A e. suc B ) )
33		nnon 4609	. . 3 |- ( B e. om -> B e. On )
34		onsucelsucr 4507	. . 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 708   = wceq 1353   e. wcel 2148   (/)c0 3422   Oncon0 4363   suc csuc 4365   omcom 4589

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-iinf 4587

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-int 3845  df-tr 4102  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590

This theorem is referenced by:  nnsucsssuc  6492  nntri3or  6493  nnsucuniel  6495  nnaordi  6508  ennnfonelemhom  12415