Theorem ordin 4913

Description: The intersection of two ordinal classes is ordinal. Proposition 7.9 of [TakeutiZaring] p. 37. (Contributed by NM, 9-May-1994.)

Assertion

Ref	Expression
ordin

Proof of Theorem ordin

Step	Hyp	Ref	Expression
1		ordtr 4897	. . 3
2		ordtr 4897	. . 3
3		trin 4555	. . 3
4	1, 2, 3	syl2an 477	. 2
5		inss2 3718	. . 3
6		trssord 4900	. . 3
7	5, 6	mp3an2 1312	. 2
8	4, 7	sylancom 667	1

Syntax hints:  ->wi 4  /\wa 369  i^icin 3474  C_wss 3475  Trwtr 4545  Ordword 4882

This theorem is referenced by:  onin  4914  ordtri3or  4915  ordelinel  4981  smores  7042  smores2  7044  ordtypelem5  7968  ordtypelem7  7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3482  df-ss 3489  df-uni 4250  df-tr 4546  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886