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Theorem onelpss 5723
 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 5692 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5692 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordelssne 5709 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
41, 2, 3syl2an 494 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1987   ≠ wne 2790   ⊆ wss 3555  Ord word 5681  Oncon0 5682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-ord 5685  df-on 5686 This theorem is referenced by:  tfindsg  7007  findsg  7040  oancom  8492  cardsdom2  8758  alephord  8842
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