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Theorem onsseleq 6405
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 6374 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6374 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordsseleq 6393 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3syl2an 596 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wss 3948  Ord word 6363  Oncon0 6364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-ord 6367  df-on 6368
This theorem is referenced by:  onsseli  6485  on0eqel  6488  onmindif2  7797  omword  8572  oeword  8592  oewordi  8593  dffi3  9428  cantnflem1d  9685  cantnflem1  9686  r1ord3g  9776  alephdom  10078  cardaleph  10086  cfsmolem  10267  ttukeylem5  10510  alephreg  10579  inar1  10772  gruina  10815  om2uzlt2i  13918  nolt02o  27205  nogt01o  27206  nosupbnd2lem1  27225  noinfbnd2lem1  27240  madebday  27402  oege2  42139  ontric3g  42355
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