Theorem ordsucss 4595

Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)

Assertion

Ref	Expression
ordsucss	|- ( Ord B -> ( A e. B -> suc A C_ B ) )

Proof of Theorem ordsucss

Step	Hyp	Ref	Expression
1		ordtr 4468	. 2 |- ( Ord B -> Tr B )
2		trss 4190	. . . . 5 |- ( Tr B -> ( A e. B -> A C_ B ) )
3		snssi 3811	. . . . . 6 |- ( A e. B -> { A } C_ B )
4	3	a1i 9	. . . . 5 |- ( Tr B -> ( A e. B -> { A } C_ B ) )
5	2, 4	jcad 307	. . . 4 |- ( Tr B -> ( A e. B -> ( A C_ B /\ { A } C_ B ) ) )
6		unss 3378	. . . 4 |- ( ( A C_ B /\ { A } C_ B ) <-> ( A u. { A } ) C_ B )
7	5, 6	imbitrdi 161	. . 3 |- ( Tr B -> ( A e. B -> ( A u. { A } ) C_ B ) )
8		df-suc 4461	. . . 4 |- suc A = ( A u. { A } )
9	8	sseq1i 3250	. . 3 |- ( suc A C_ B <-> ( A u. { A } ) C_ B )
10	7, 9	imbitrrdi 162	. 2 |- ( Tr B -> ( A e. B -> suc A C_ B ) )
11	1, 10	syl 14	1 |- ( Ord B -> ( A e. B -> suc A C_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   u. cun 3195   C_ wss 3197   {csn 3666   Tr wtr 4181   Ord word 4452   suc csuc 4455

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-uni 3888  df-tr 4182  df-iord 4456  df-suc 4461

This theorem is referenced by:  ordelsuc  4596  tfrlemibfn  6472  tfr1onlembfn  6488  tfrcllembfn  6501  sucinc2  6590  nndomo  7021  prarloclemn  7682  ennnfonelemhom  12981  ennnfonelemrn  12985