Theorem ordsucss 4552

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 4425	. 2 |- ( Ord B -> Tr B )
2		trss 4151	. . . . 5 |- ( Tr B -> ( A e. B -> A C_ B ) )
3		snssi 3777	. . . . . 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 3347	. . . 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 4418	. . . 4 |- suc A = ( A u. { A } )
9	8	sseq1i 3219	. . 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 2176   u. cun 3164   C_ wss 3166   {csn 3633   Tr wtr 4142   Ord word 4409   suc csuc 4412

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-uni 3851  df-tr 4143  df-iord 4413  df-suc 4418

This theorem is referenced by:  ordelsuc  4553  tfrlemibfn  6414  tfr1onlembfn  6430  tfrcllembfn  6443  sucinc2  6532  nndomo  6961  prarloclemn  7612  ennnfonelemhom  12786  ennnfonelemrn  12790