Theorem nnsucelsuc 6768

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4657, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4679. (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 2302	. . . 4 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
2		suceq 4549	. . . . 5 |- ( x = (/) -> suc x = suc (/) )
3	2	eleq2d 2308	. . . 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 2302	. . . 4 |- ( x = y -> ( A e. x <-> A e. y ) )
6		suceq 4549	. . . . 5 |- ( x = y -> suc x = suc y )
7	6	eleq2d 2308	. . . 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 2302	. . . 4 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
10		suceq 4549	. . . . 5 |- ( x = suc y -> suc x = suc suc y )
11	10	eleq2d 2308	. . . 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 2302	. . . 4 |- ( x = B -> ( A e. x <-> A e. B ) )
14		suceq 4549	. . . . 5 |- ( x = B -> suc x = suc B )
15	14	eleq2d 2308	. . . 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 3525	. . . 4 |- -. A e. (/)
18	17	pm2.21i 655	. . 3 |- ( A e. (/) -> suc A e. suc (/) )
19		elsuci 4550	. . . . . . . 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 4549	. . . . . . . . 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 798	. . . . . . 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 2824	. . . . . . . 8 |- y e. _V
27	26	sucex 4648	. . . . . . 7 |- suc y e. _V
28	27	elsuc2 4554	. . . . . 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 4749	. 2 |- ( B e. om -> ( A e. B -> suc A e. suc B ) )
33		nnon 4759	. . 3 |- ( B e. om -> B e. On )
34		onsucelsucr 4657	. . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   e. wcel 2209   (/)c0 3520   Oncon0 4510   suc csuc 4512   omcom 4739

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-tr 4232  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740

This theorem is used by:  nnsucsssuc  6769  nntri3or  6770  nnsucuniel  6772  nnaordi  6785  ennnfonelemhom  13364