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Theorem onpsssuc 7816
Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Assertion
Ref Expression
onpsssuc (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)

Proof of Theorem onpsssuc
StepHypRef Expression
1 sucidg 6444 . 2 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6373 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordsuc 7810 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
42, 3sylib 217 . . 3 (𝐴 ∈ On → Ord suc 𝐴)
5 ordelpss 6391 . . 3 ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
62, 4, 5syl2anc 583 . 2 (𝐴 ∈ On → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
71, 6mpbid 231 1 (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2099  wpss 3945  Ord word 6362  Oncon0 6363  suc csuc 6365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-suc 6369
This theorem is referenced by:  ackbij1lem15  10249
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