Theorem onsucelsucr 4556

Description: Membership is inherited by predecessors. The converse, for all ordinals, implies excluded middle, as shown at onsucelsucexmid 4578. However, the converse does hold where B is a natural number, as seen at nnsucelsuc 6577. (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 2783	. . . 4 |- ( suc A e. suc B -> suc A e. _V )
2		sucexb 4545	. . . 4 |- ( A e. _V <-> suc A e. _V )
3	1, 2	sylibr 134	. . 3 |- ( suc A e. suc B -> A e. _V )
4		onelss 4434	. . . . . . 7 |- ( B e. On -> ( suc A e. B -> suc A C_ B ) )
5		eqimss 3247	. . . . . . . 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 719	. . . . . 6 |- ( B e. On -> ( ( suc A e. B \/ suc A = B ) -> suc A C_ B ) )
8	7	adantl 277	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( ( suc A e. B \/ suc A = B ) -> suc A C_ B ) )
9		elsucg 4451	. . . . . . 7 |- ( suc A e. _V -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
10	2, 9	sylbi 121	. . . . . 6 |- ( A e. _V -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
11	10	adantr 276	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( suc A e. suc B <-> ( suc A e. B \/ suc A = B ) ) )
12		eloni 4422	. . . . . 6 |- ( B e. On -> Ord B )
13		ordelsuc 4553	. . . . . 6 |- ( ( A e. _V /\ Ord B ) -> ( A e. B <-> suc A C_ B ) )
14	12, 13	sylan2 286	. . . . 5 |- ( ( A e. _V /\ B e. On ) -> ( A e. B <-> suc A C_ B ) )
15	8, 11, 14	3imtr4d 203	. . . 4 |- ( ( A e. _V /\ B e. On ) -> ( suc A e. suc B -> A e. B ) )
16	15	impancom 260	. . 3 |- ( ( A e. _V /\ suc A e. suc B ) -> ( B e. On -> A e. B ) )
17	3, 16	mpancom 422	. 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 104   <-> wb 105   \/ wo 710   = wceq 1373   e. wcel 2176   _Vcvv 2772   C_ wss 3166   Ord word 4409   Oncon0 4410   suc csuc 4412

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-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 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  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-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418

This theorem is referenced by:  nnsucelsuc  6577