Theorem sucinc2 6026

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 4112	. . . . 5 |- ( B e. On -> Ord B )
2		ordsucss 4230	. . . . 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 115	. . 3 |- ( ( B e. On /\ A e. B ) -> suc A C_ B )
5		sssucid 4152	. . 3 |- B C_ suc B
6	4, 5	syl6ss 2957	. 2 |- ( ( B e. On /\ A e. B ) -> suc A C_ suc B )
7		onelon 4121	. . 3 |- ( ( B e. On /\ A e. B ) -> A e. On )
8		elex 2566	. . . 4 |- ( A e. On -> A e. _V )
9		sucexg 4224	. . . 4 |- ( A e. On -> suc A e. _V )
10		suceq 4139	. . . . 5 |- ( z = A -> suc z = suc A )
11		sucinc.1	. . . . 5 |- F = ( z e. _V |-> suc z )
12	10, 11	fvmptg 5248	. . . 4 |- ( ( A e. _V /\ suc A e. _V ) -> ( F ` A ) = suc A )
13	8, 9, 12	syl2anc 391	. . 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 2566	. . . 4 |- ( B e. On -> B e. _V )
16		sucexg 4224	. . . 4 |- ( B e. On -> suc B e. _V )
17		suceq 4139	. . . . 5 |- ( z = B -> suc z = suc B )
18	17, 11	fvmptg 5248	. . . 4 |- ( ( B e. _V /\ suc B e. _V ) -> ( F ` B ) = suc B )
19	15, 16, 18	syl2anc 391	. . 3 |- ( B e. On -> ( F ` B ) = suc B )
20	19	adantr 261	. 2 |- ( ( B e. On /\ A e. B ) -> ( F ` B ) = suc B )
21	6, 14, 20	3sstr4d 2988	1 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   C_ wss 2917   |-> cmpt 3818   Ord word 4099   Oncon0 4100   suc csuc 4102   ` cfv 4902

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-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910

This theorem is referenced by: (None)