Theorem onsucelsucr 4234

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4255. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6070. (Contributed by Jim Kingdon, 17-Jul-2019.)

Assertion

Ref	Expression
onsucelsucr	|- ( B e. On -> ( suc A e. suc B -> A e. B ) )

Proof of Theorem onsucelsucr

Step	Hyp	Ref	Expression
1		elex 2566	. . . 4 |- ( suc A e. suc B -> suc A e. _V )
2		sucexb 4223	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 137	. . 3 |- ( suc A e. suc B -> A e. _V )
4		onelss 4124	. . . . . . 7 |- ( B e. On -> ( suc A e. B -> suc A C_ B ) )
5		eqimss 2997	. . . . . . . 8 |- ( suc A = B -> suc A C_ B )
6	5	a1i 9	. . . . . . 7 |- ( B e. On -> ( suc A = B -> suc A C_ B ) )
7	4, 6	jaod 637	. . . . . 6 |- ( B e. On -> ( ( suc A e. B \/ suc A = B ) -> suc A C_ B ) )
8	7	adantl 262	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( ( suc A e. B \/ suc A = B ) -> suc A C_ B ) )
9		elsucg 4141	. . . . . . 7 |- ( suc A e. _V -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
10	2, 9	sylbi 114	. . . . . 6 |- ( A e. _V -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
11	10	adantr 261	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
12		eloni 4112	. . . . . 6 |- ( B e. On -> Ord B )
13		ordelsuc 4231	. . . . . 6 |- ( ( A e. _V /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
14	12, 13	sylan2 270	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( A e. B <-> suc A C_ B ) )
15	8, 11, 14	3imtr4d 192	. . . 4 |- ( ( A e. _V /\ B e. On ) -> ( suc A e. suc B -> A e. B ) )
16	15	impancom 247	. . 3 |- ( ( A e. _V /\ suc A e. suc B ) -> ( B e. On -> A e. B ) )
17	3, 16	mpancom 399	. 2 |- ( suc A e. suc B -> ( B e. On -> A e. B ) )
18	17	com12 27	1 |- ( B e. On -> ( suc A e. suc B -> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   Ord word 4099   Oncon0 4100   suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by:  nnsucelsuc  6070