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Theorem onelpss 6004
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 5974 . 2 (𝐴 ∈ On → Ord 𝐴)
2 eloni 5974 . 2 (𝐵 ∈ On → Ord 𝐵)
3 ordelssne 5991 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
41, 2, 3syl2an 591 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wcel 2166  wne 3000  wss 3799  Ord word 5963  Oncon0 5964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-tr 4977  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-ord 5967  df-on 5968
This theorem is referenced by:  tfindsg  7322  findsg  7355  oancom  8826  cardsdom2  9128  alephord  9212  scutbdaylt  32462
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