Theorem sucinc2 6613

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 4472	. . . . 5 |- ( B e. On -> Ord B )
2		ordsucss 4602	. . . . 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 124	. . 3 |- ( ( B e. On /\ A e. B ) -> suc A C_ B )
5		sssucid 4512	. . 3 |- B C_ suc B
6	4, 5	sstrdi 3239	. 2 |- ( ( B e. On /\ A e. B ) -> suc A C_ suc B )
7		onelon 4481	. . 3 |- ( ( B e. On /\ A e. B ) -> A e. On )
8		elex 2814	. . . 4 |- ( A e. On -> A e. _V )
9		sucexg 4596	. . . 4 |- ( A e. On -> suc A e. _V )
10		suceq 4499	. . . . 5 |- ( z = A -> suc z = suc A )
11		sucinc.1	. . . . 5 |- F = ( z e. _V |-> suc z )
12	10, 11	fvmptg 5722	. . . 4 |- ( ( A e. _V /\ suc A e. _V ) -> ( F ` A ) = suc A )
13	8, 9, 12	syl2anc 411	. . 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 2814	. . . 4 |- ( B e. On -> B e. _V )
16		sucexg 4596	. . . 4 |- ( B e. On -> suc B e. _V )
17		suceq 4499	. . . . 5 |- ( z = B -> suc z = suc B )
18	17, 11	fvmptg 5722	. . . 4 |- ( ( B e. _V /\ suc B e. _V ) -> ( F ` B ) = suc B )
19	15, 16, 18	syl2anc 411	. . 3 |- ( B e. On -> ( F ` B ) = suc B )
20	19	adantr 276	. 2 |- ( ( B e. On /\ A e. B ) -> ( F ` B ) = suc B )
21	6, 14, 20	3sstr4d 3272	1 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   |-> cmpt 4150   Ord word 4459   Oncon0 4460   suc csuc 4462   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334

This theorem is referenced by: (None)