Theorem onsucelsucr 4485

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4507. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6459. (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 2737	. . . 4 |- ( suc A e. suc B -> suc A e. _V )
2		sucexb 4474	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 133	. . 3 |- ( suc A e. suc B -> A e. _V )
4		onelss 4365	. . . . . . 7 |- ( B e. On -> ( suc A e. B -> suc A C_ B ) )
5		eqimss 3196	. . . . . . . 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 707	. . . . . 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 4382	. . . . . . 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 4353	. . . . . 6 |- ( B e. On -> Ord B )
13		ordelsuc 4482	. . . . . 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 419	. 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 698   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   Ord word 4340   Oncon0 4341   suc csuc 4343

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  nnsucelsuc  6459