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Theorem onelpss 6411
Description: Relationship between membership and proper subset of an ordinal number. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
onelpss ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))

Proof of Theorem onelpss
StepHypRef Expression
1 eloni 6381 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 6381 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordelssne 6398 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
41, 2, 3syl2an 594 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2098  wne 2929  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-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:  tfindsg  7866  findsg  7905  oancom  9676  cardsdom2  10013  alephord  10100  scutbdaylt  27797  omabs2  42903  naddwordnexlem4  42973  omssrncard  43112
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