Theorem ordsucss 4608

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 4481	. 2 |- ( Ord B -> Tr B )
2		trss 4201	. . . . 5 |- ( Tr B -> ( A e. B -> A C_ B ) )
3		snssi 3822	. . . . . 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 3383	. . . 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 4474	. . . 4 |- suc A = ( A u. { A } )
9	8	sseq1i 3254	. . 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 2202   u. cun 3199   C_ wss 3201   {csn 3673   Tr wtr 4192   Ord word 4465   suc csuc 4468

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-uni 3899  df-tr 4193  df-iord 4469  df-suc 4474

This theorem is referenced by:  ordelsuc  4609  tfrlemibfn  6537  tfr1onlembfn  6553  tfrcllembfn  6566  sucinc2  6657  nndomo  7093  prarloclemn  7762  ennnfonelemhom  13099  ennnfonelemrn  13103