Theorem nnsucelsuc 6482

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4501, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4523. (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 2239	. . . 4 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
2		suceq 4396	. . . . 5 |- ( x = (/) -> suc x = suc (/) )
3	2	eleq2d 2245	. . . 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 2239	. . . 4 |- ( x = y -> ( A e. x <-> A e. y ) )
6		suceq 4396	. . . . 5 |- ( x = y -> suc x = suc y )
7	6	eleq2d 2245	. . . 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 2239	. . . 4 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
10		suceq 4396	. . . . 5 |- ( x = suc y -> suc x = suc suc y )
11	10	eleq2d 2245	. . . 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 2239	. . . 4 |- ( x = B -> ( A e. x <-> A e. B ) )
14		suceq 4396	. . . . 5 |- ( x = B -> suc x = suc B )
15	14	eleq2d 2245	. . . 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 3424	. . . 4 |- -. A e. (/)
18	17	pm2.21i 646	. . 3 |- ( A e. (/) -> suc A e. suc (/) )
19		elsuci 4397	. . . . . . . 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 4396	. . . . . . . . 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 2738	. . . . . . . 8 |- y e. _V
27	26	sucex 4492	. . . . . . 7 |- suc y e. _V
28	27	elsuc2 4401	. . . . . 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 4593	. 2 |- ( B e. om -> ( A e. B -> suc A e. suc B ) )
33		nnon 4603	. . 3 |- ( B e. om -> B e. On )
34		onsucelsucr 4501	. . 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 2146   (/)c0 3420   Oncon0 4357   suc csuc 4359   omcom 4583

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-iinf 4581

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-uni 3806  df-int 3841  df-tr 4097  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584

This theorem is referenced by:  nnsucsssuc  6483  nntri3or  6484  nnsucuniel  6486  nnaordi  6499  ennnfonelemhom  12383