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Theorem onpsssuc 7519
 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 6247 . 2 (𝐴 ∈ On → 𝐴 ∈ suc 𝐴)
2 eloni 6179 . . 3 (𝐴 ∈ On → Ord 𝐴)
3 ordsuc 7514 . . . 4 (Ord 𝐴 ↔ Ord suc 𝐴)
42, 3sylib 221 . . 3 (𝐴 ∈ On → Ord suc 𝐴)
5 ordelpss 6197 . . 3 ((Ord 𝐴 ∧ Ord suc 𝐴) → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
62, 4, 5syl2anc 587 . 2 (𝐴 ∈ On → (𝐴 ∈ suc 𝐴𝐴 ⊊ suc 𝐴))
71, 6mpbid 235 1 (𝐴 ∈ On → 𝐴 ⊊ suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∈ wcel 2114   ⊊ wpss 3909  Ord word 6168  Oncon0 6169  suc csuc 6171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-ord 6172  df-on 6173  df-suc 6175 This theorem is referenced by:  ackbij1lem15  9645
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