ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  onpsssuc Structured version   GIF version

Theorem onpsssuc 4247
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 4224 . . . 4 ¬ A A
2 sucidg 4119 . . . . 5 (A On → A suc A)
3 eleq2 2098 . . . . 5 (A = suc A → (A AA suc A))
42, 3syl5ibrcom 146 . . . 4 (A On → (A = suc AA A))
51, 4mtoi 589 . . 3 (A On → ¬ A = suc A)
6 sssucid 4118 . . 3 A ⊆ suc A
75, 6jctil 295 . 2 (A On → (A ⊆ suc A ¬ A = suc A))
8 dfpss2 3023 . 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 1242   wcel 1390  wss 2911  wpss 2912  Oncon0 4066  suc csuc 4068
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pss 2927  df-sn 3373  df-suc 4074
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator