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Theorem onsseleq 6017
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 5986 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5986 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordsseleq 6005 . 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 836   = wceq 1601  wcel 2106  wss 3791  Ord word 5975  Oncon0 5976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-tr 4988  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-ord 5979  df-on 5980
This theorem is referenced by:  onsseli  6090  on0eqel  6093  onmindif2  7290  omword  7934  oeword  7954  oewordi  7955  dffi3  8625  cantnflem1d  8882  cantnflem1  8883  r1ord3g  8939  alephdom  9237  cardaleph  9245  cfsmolem  9427  ttukeylem5  9670  alephreg  9739  inar1  9932  gruina  9975  om2uzlt2i  13069  nolt02o  32434  nosupbnd2lem1  32450
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