Theorem onsucelsucr 4492

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4514. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6470. (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 2741	. . . 4 |- ( suc A e. suc B -> suc A e. _V )
2		sucexb 4481	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 133	. . 3 |- ( suc A e. suc B -> A e. _V )
4		onelss 4372	. . . . . . 7 |- ( B e. On -> ( suc A e. B -> suc A C_ B ) )
5		eqimss 3201	. . . . . . . 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 712	. . . . . 6 |- ( B e. On -> ( ( suc A e. B \/ suc A = B ) -> suc A C_ B ) )
8	7	adantl 275	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( ( suc A e. B \/ suc A = B ) -> suc A C_ B ) )
9		elsucg 4389	. . . . . . 7 |- ( suc A e. _V -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
10	2, 9	sylbi 120	. . . . . 6 |- ( A e. _V -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
11	10	adantr 274	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
12		eloni 4360	. . . . . 6 |- ( B e. On -> Ord B )
13		ordelsuc 4489	. . . . . 6 |- ( ( A e. _V /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
14	12, 13	sylan2 284	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( A e. B <-> suc A C_ B ) )
15	8, 11, 14	3imtr4d 202	. . . 4 |- ( ( A e. _V /\ B e. On ) -> ( suc A e. suc B -> A e. B ) )
16	15	impancom 258	. . 3 |- ( ( A e. _V /\ suc A e. suc B ) -> ( B e. On -> A e. B ) )
17	3, 16	mpancom 420	. 2 |- ( suc A e. suc B -> ( B e. On -> A e. B ) )
18	17	com12 30	1 |- ( B e. On -> ( suc A e. suc B -> A e. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   Ord word 4347   Oncon0 4348   suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by:  nnsucelsuc  6470