Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordsson Structured version   Visualization version   GIF version

Theorem ordsson 6936
 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 𝐴𝐴 ⊆ On)

Proof of Theorem ordsson
StepHypRef Expression
1 ordon 6929 . 2 Ord On
2 ordeleqon 6935 . . . . 5 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
32biimpi 206 . . . 4 (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
43adantr 481 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On))
5 ordsseleq 5711 . . 3 ((Ord 𝐴 ∧ Ord On) → (𝐴 ⊆ On ↔ (𝐴 ∈ On ∨ 𝐴 = On)))
64, 5mpbird 247 . 2 ((Ord 𝐴 ∧ Ord On) → 𝐴 ⊆ On)
71, 6mpan2 706 1 (Ord 𝐴𝐴 ⊆ On)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1480   ∈ wcel 1987   ⊆ wss 3555  Ord word 5681  Oncon0 5682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686 This theorem is referenced by:  onss  6937  orduni  6941  ordsucuniel  6971  ordsucuni  6976  iordsmo  7399  dfrecs3  7414  tfr2b  7437  tz7.44-2  7448  ordiso2  8364  ordtypelem7  8373  ordtypelem8  8374  oiid  8390  r1tr  8583  r1ordg  8585  r1ord3g  8586  r1pwss  8591  r1val1  8593  rankwflemb  8600  r1elwf  8603  rankr1ai  8605  cflim2  9029  cfss  9031  cfslb  9032  cfslbn  9033  cfslb2n  9034  cofsmo  9035  coftr  9039  inaprc  9602  dford5  31314  rdgprc  31398  nosepon  31520  limsucncmpi  32083
 Copyright terms: Public domain W3C validator