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Theorem onpsssuc 4231
 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 elirr 4208 . . . 4 ¬ A A
2 sucidg 4102 . . . . 5 (A On → A suc A)
3 eleq2 2083 . . . . 5 (A = suc A → (A AA suc A))
42, 3syl5ibrcom 146 . . . 4 (A On → (A = suc AA A))
51, 4mtoi 577 . . 3 (A On → ¬ A = suc A)
6 sssucid 4101 . . 3 A ⊆ suc A
75, 6jctil 295 . 2 (A On → (A ⊆ suc A ¬ A = suc A))
8 dfpss2 3006 . 2 (A ⊊ suc A ↔ (A ⊆ suc A ¬ A = suc A))
97, 8sylibr 137 1 (A On → A ⊊ suc A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1228   ∈ wcel 1374   ⊆ wss 2894   ⊊ wpss 2895  Oncon0 4049  suc csuc 4051 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-pss 2910  df-sn 3356  df-suc 4057 This theorem is referenced by: (None)
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