Theorem sucinc2 6557

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 4441	. . . . 5 |- ( B e. On -> Ord B )
2		ordsucss 4571	. . . . 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 4481	. . 3 |- B C_ suc B
6	4, 5	sstrdi 3214	. 2 |- ( ( B e. On /\ A e. B ) -> suc A C_ suc B )
7		onelon 4450	. . 3 |- ( ( B e. On /\ A e. B ) -> A e. On )
8		elex 2789	. . . 4 |- ( A e. On -> A e. _V )
9		sucexg 4565	. . . 4 |- ( A e. On -> suc A e. _V )
10		suceq 4468	. . . . 5 |- ( z = A -> suc z = suc A )
11		sucinc.1	. . . . 5 |- F = ( z e. _V |-> suc z )
12	10, 11	fvmptg 5680	. . . 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 2789	. . . 4 |- ( B e. On -> B e. _V )
16		sucexg 4565	. . . 4 |- ( B e. On -> suc B e. _V )
17		suceq 4468	. . . . 5 |- ( z = B -> suc z = suc B )
18	17, 11	fvmptg 5680	. . . 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 3247	1 |- ( ( B e. On /\ A e. B ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2777   C_ wss 3175   |-> cmpt 4122   Ord word 4428   Oncon0 4429   suc csuc 4431   ` cfv 5291

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2779  df-sbc 3007  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-br 4061  df-opab 4123  df-mpt 4124  df-tr 4160  df-id 4359  df-iord 4432  df-on 4434  df-suc 4437  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-iota 5252  df-fun 5293  df-fv 5299

This theorem is referenced by: (None)