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

Theorem onsseleq 6008
Description: Relationship between subset and membership of an ordinal number. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onsseleq ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))

Proof of Theorem onsseleq
StepHypRef Expression
1 eloni 5977 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5977 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordsseleq 5996 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3syl2an 589 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wo 878   = wceq 1656  wcel 2164  wss 3798  Ord word 5966  Oncon0 5967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-tr 4978  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-ord 5970  df-on 5971
This theorem is referenced by:  onsseli  6081  on0eqel  6084  onmindif2  7278  omword  7922  oeword  7942  oewordi  7943  dffi3  8612  cantnflem1d  8869  cantnflem1  8870  r1ord3g  8926  alephdom  9224  cardaleph  9232  cfsmolem  9414  ttukeylem5  9657  alephreg  9726  inar1  9919  gruina  9962  om2uzlt2i  13052  nolt02o  32379  nosupbnd2lem1  32395
  Copyright terms: Public domain W3C validator