Theorem sucinc2 6350

Description: Successor is increasing. (Contributed by Jim Kingdon, 14-Jul-2019.)

Hypothesis

Ref	Expression
sucinc.1	|- F = ( z e. _V |-> suc z )

Assertion

Ref	Expression
sucinc2	|- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Distinct variable groups:   z, A   z, B

Allowed substitution hint:   F( z)

Proof of Theorem sucinc2

Step	Hyp	Ref	Expression
1		eloni 4305	. . . . 5 |- ( B e. On -> Ord B )
2		ordsucss 4428	. . . . 5 |- ( Ord B -> ( A e. B -> suc A C_ B ) )
3	1, 2	syl 14	. . . 4 |- ( B e. On -> ( A e. B -> suc A C_ B ) )
4	3	imp 123	. . 3 |- ( ( B e. On /\ A e. B ) -> suc A C_ B )
5		sssucid 4345	. . 3 |- B C_ suc B
6	4, 5	sstrdi 3114	. 2 |- ( ( B e. On /\ A e. B ) -> suc A C_ suc B )
7		onelon 4314	. . 3 |- ( ( B e. On /\ A e. B ) -> A e. On )
8		elex 2700	. . . 4 |- ( A e. On -> A e. _V )
9		sucexg 4422	. . . 4 |- ( A e. On -> suc A e. _V )
10		suceq 4332	. . . . 5 |- ( z = A -> suc z = suc A )
11		sucinc.1	. . . . 5 |- F = ( z e. _V |-> suc z )
12	10, 11	fvmptg 5505	. . . 4 |- ( ( A e. _V /\ suc A e. _V ) -> ( F ` A ) = suc A )
13	8, 9, 12	syl2anc 409	. . 3 |- ( A e. On -> ( F ` A ) = suc A )
14	7, 13	syl 14	. 2 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) = suc A )
15		elex 2700	. . . 4 |- ( B e. On -> B e. _V )
16		sucexg 4422	. . . 4 |- ( B e. On -> suc B e. _V )
17		suceq 4332	. . . . 5 |- ( z = B -> suc z = suc B )
18	17, 11	fvmptg 5505	. . . 4 |- ( ( B e. _V /\ suc B e. _V ) -> ( F ` B ) = suc B )
19	15, 16, 18	syl2anc 409	. . 3 |- ( B e. On -> ( F ` B ) = suc B )
20	19	adantr 274	. 2 |- ( ( B e. On /\ A e. B ) -> ( F ` B ) = suc B )
21	6, 14, 20	3sstr4d 3147	1 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   C_ wss 3076   |-> cmpt 3997   Ord word 4292   Oncon0 4293   suc csuc 4295   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139

This theorem is referenced by: (None)