Theorem nnsucelsuc 6579

Description: Membership is inherited by successors. The reverse direction holds for all ordinals, as seen at onsucelsucr 4557, but the forward direction, for all ordinals, implies excluded middle as seen as onsucelsucexmid 4579. (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 2269	. . . 4 |- ( x = (/) -> ( A e. x <-> A e. (/) ) )
2		suceq 4450	. . . . 5 |- ( x = (/) -> suc x = suc (/) )
3	2	eleq2d 2275	. . . 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 2269	. . . 4 |- ( x = y -> ( A e. x <-> A e. y ) )
6		suceq 4450	. . . . 5 |- ( x = y -> suc x = suc y )
7	6	eleq2d 2275	. . . 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 2269	. . . 4 |- ( x = suc y -> ( A e. x <-> A e. suc y ) )
10		suceq 4450	. . . . 5 |- ( x = suc y -> suc x = suc suc y )
11	10	eleq2d 2275	. . . 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 2269	. . . 4 |- ( x = B -> ( A e. x <-> A e. B ) )
14		suceq 4450	. . . . 5 |- ( x = B -> suc x = suc B )
15	14	eleq2d 2275	. . . 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 3464	. . . 4 |- -. A e. (/)
18	17	pm2.21i 647	. . 3 |- ( A e. (/) -> suc A e. suc (/) )
19		elsuci 4451	. . . . . . . 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 4450	. . . . . . . . 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 788	. . . . . . 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 2775	. . . . . . . 8 |- y e. _V
27	26	sucex 4548	. . . . . . 7 |- suc y e. _V
28	27	elsuc2 4455	. . . . . 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 4649	. 2 |- ( B e. om -> ( A e. B -> suc A e. suc B ) )
33		nnon 4659	. . 3 |- ( B e. om -> B e. On )
34		onsucelsucr 4557	. . 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 710   = wceq 1373   e. wcel 2176   (/)c0 3460   Oncon0 4411   suc csuc 4413   omcom 4639

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-iinf 4637

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-tr 4144  df-iord 4414  df-on 4416  df-suc 4419  df-iom 4640

This theorem is referenced by:  nnsucsssuc  6580  nntri3or  6581  nnsucuniel  6583  nnaordi  6596  ennnfonelemhom  12819