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Theorem onpsssuc 7061
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 5841 . 2 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 5771 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordsuc 7056 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
42, 3sylib 208 . . 3 (𝐴 ∈ On → Ord suc 𝐴)
5 ordelpss 5789 . . 3 ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
62, 4, 5syl2anc 694 . 2 (𝐴 ∈ On → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
71, 6mpbid 222 1 (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2030  wpss 3608  Ord word 5760  Oncon0 5761  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765  df-suc 5767
This theorem is referenced by:  ackbij1lem15  9094
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