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Theorem onpsssuc 7804
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 6439 . 2 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6368 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordsuc 7798 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
42, 3sylib 217 . . 3 (𝐴 ∈ On → Ord suc 𝐴)
5 ordelpss 6386 . . 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 2098  wpss 3944  Ord word 6357  Oncon0 6358  suc csuc 6360
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362  df-suc 6364
This theorem is referenced by:  ackbij1lem15  10231
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