Theorem sucinc2 6049

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 4130	. . . . 5 |- ( B e. On -> Ord B )
2		ordsucss 4248	. . . . 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 122	. . 3 |- ( ( B e. On /\ A e. B ) -> suc A C_ B )
5		sssucid 4170	. . 3 |- B C_ suc B
6	4, 5	syl6ss 3011	. 2 |- ( ( B e. On /\ A e. B ) -> suc A C_ suc B )
7		onelon 4139	. . 3 |- ( ( B e. On /\ A e. B ) -> A e. On )
8		elex 2610	. . . 4 |- ( A e. On -> A e. _V )
9		sucexg 4242	. . . 4 |- ( A e. On -> suc A e. _V )
10		suceq 4157	. . . . 5 |- ( z = A -> suc z = suc A )
11		sucinc.1	. . . . 5 |- F = ( z e. _V |-> suc z )
12	10, 11	fvmptg 5269	. . . 4 |- ( ( A e. _V /\ suc A e. _V ) -> ( F ` A ) = suc A )
13	8, 9, 12	syl2anc 403	. . 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 2610	. . . 4 |- ( B e. On -> B e. _V )
16		sucexg 4242	. . . 4 |- ( B e. On -> suc B e. _V )
17		suceq 4157	. . . . 5 |- ( z = B -> suc z = suc B )
18	17, 11	fvmptg 5269	. . . 4 |- ( ( B e. _V /\ suc B e. _V ) -> ( F ` B ) = suc B )
19	15, 16, 18	syl2anc 403	. . 3 |- ( B e. On -> ( F ` B ) = suc B )
20	19	adantr 270	. 2 |- ( ( B e. On /\ A e. B ) -> ( F ` B ) = suc B )
21	6, 14, 20	3sstr4d 3042	1 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   C_ wss 2973   |-> cmpt 3839   Ord word 4117   Oncon0 4118   suc csuc 4120   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by: (None)