Theorem ordsucss 4551

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 4424	. 2 |- ( Ord B -> Tr B )
2		trss 4150	. . . . 5 |- ( Tr B -> ( A e. B -> A C_ B ) )
3		snssi 3776	. . . . . 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 3346	. . . 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 4417	. . . 4 |- suc A = ( A u. { A } )
9	8	sseq1i 3218	. . 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 2175   u. cun 3163   C_ wss 3165   {csn 3632   Tr wtr 4141   Ord word 4408   suc csuc 4411

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-sn 3638  df-uni 3850  df-tr 4142  df-iord 4412  df-suc 4417

This theorem is referenced by:  ordelsuc  4552  tfrlemibfn  6413  tfr1onlembfn  6429  tfrcllembfn  6442  sucinc2  6531  nndomo  6960  prarloclemn  7611  ennnfonelemhom  12757  ennnfonelemrn  12761