Theorem sucinc2 6414

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 4353	. . . . 5 |- ( B e. On -> Ord B )
2		ordsucss 4481	. . . . 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 4393	. . 3 |- B C_ suc B
6	4, 5	sstrdi 3154	. 2 |- ( ( B e. On /\ A e. B ) -> suc A C_ suc B )
7		onelon 4362	. . 3 |- ( ( B e. On /\ A e. B ) -> A e. On )
8		elex 2737	. . . 4 |- ( A e. On -> A e. _V )
9		sucexg 4475	. . . 4 |- ( A e. On -> suc A e. _V )
10		suceq 4380	. . . . 5 |- ( z = A -> suc z = suc A )
11		sucinc.1	. . . . 5 |- F = ( z e. _V |-> suc z )
12	10, 11	fvmptg 5562	. . . 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 2737	. . . 4 |- ( B e. On -> B e. _V )
16		sucexg 4475	. . . 4 |- ( B e. On -> suc B e. _V )
17		suceq 4380	. . . . 5 |- ( z = B -> suc z = suc B )
18	17, 11	fvmptg 5562	. . . 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 3187	1 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   |-> cmpt 4043   Ord word 4340   Oncon0 4341   suc csuc 4343   ` cfv 5188

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-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by: (None)