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Theorem onpsssuc 7830
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 6455 . 2 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6384 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordsuc 7824 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
42, 3sylib 217 . . 3 (𝐴 ∈ On → Ord suc 𝐴)
5 ordelpss 6402 . . 3 ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
62, 4, 5syl2anc 582 . 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 3950  Ord word 6373  Oncon0 6374  suc csuc 6376
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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378  df-suc 6380
This theorem is referenced by:  ackbij1lem15  10267
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