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Theorem onsseleq 6412
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 6381 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6381 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordsseleq 6400 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
41, 2, 3syl2an 594 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  wcel 2098  wss 3944  Ord word 6370  Oncon0 6371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375
This theorem is referenced by:  onsseli  6492  on0eqel  6495  onmindif2  7811  omword  8591  oeword  8611  oewordi  8612  dffi3  9456  cantnflem1d  9713  cantnflem1  9714  r1ord3g  9804  alephdom  10106  cardaleph  10114  cfsmolem  10295  ttukeylem5  10538  alephreg  10607  inar1  10800  gruina  10843  om2uzlt2i  13952  nolt02o  27674  nogt01o  27675  nosupbnd2lem1  27694  noinfbnd2lem1  27709  madebday  27872  om2noseqlt2  28223  oege2  42878  ontric3g  43094
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