Theorem ordsson 6989

Description: Any ordinal class is a subclass of the class of ordinal numbers. Corollary 7.15 of [TakeutiZaring] p. 38. (Contributed by NM, 18-May-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
ordsson	|- ( Ord A -> A C_ On )

Proof of Theorem ordsson

Step	Hyp	Ref	Expression
1		ordon 6982	. 2 |- Ord On
2		ordeleqon 6988	. . . . 5 |- ( Ord A <-> ( A e. On \/ A = On ) )
3	2	biimpi 206	. . . 4 |- ( Ord A -> ( A e. On \/ A = On ) )
4	3	adantr 481	. . 3 |- ( ( Ord A /\ Ord On ) -> ( A e. On \/ A = On ) )
5		ordsseleq 5752	. . 3 |- ( ( Ord A /\ Ord On ) -> ( A C_ On <-> ( A e. On \/ A = On ) ) )
6	4, 5	mpbird 247	. 2 |- ( ( Ord A /\ Ord On ) -> A C_ On )
7	1, 6	mpan2 707	1 |- ( Ord A -> A C_ On )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   Ord word 5722   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  onss  6990  orduni  6994  ordsucuniel  7024  ordsucuni  7029  iordsmo  7454  dfrecs3  7469  tfr2b  7492  tz7.44-2  7503  ordiso2  8420  ordtypelem7  8429  ordtypelem8  8430  oiid  8446  r1tr  8639  r1ordg  8641  r1ord3g  8642  r1pwss  8647  r1val1  8649  rankwflemb  8656  r1elwf  8659  rankr1ai  8661  cflim2  9085  cfss  9087  cfslb  9088  cfslbn  9089  cfslb2n  9090  cofsmo  9091  coftr  9095  inaprc  9658  dford5  31608  rdgprc  31700  nosepon  31818  limsucncmpi  32444