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

Theorem onsseleq 5753
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 5721 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5721 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordsseleq 5740 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3syl2an 494 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wcel 1988  wss 3567  Ord word 5710  Oncon0 5711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-ord 5714  df-on 5715
This theorem is referenced by:  onsseli  5830  on0eqel  5833  onmindif2  6997  omword  7635  oeword  7655  oewordi  7656  dffi3  8322  cantnflem1d  8570  cantnflem1  8571  r1ord3g  8627  alephdom  8889  cardaleph  8897  cfsmolem  9077  ttukeylem5  9320  alephreg  9389  inar1  9582  gruina  9625  om2uzlt2i  12733  nolt02o  31819  nosupbnd2lem1  31835
  Copyright terms: Public domain W3C validator