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Theorem sucprcreg 4227
 Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)
Assertion
Ref Expression
sucprcreg A V ↔ suc A = A)

Proof of Theorem sucprcreg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 sucprc 4115 . 2 A V → suc A = A)
2 elirr 4224 . . . 4 ¬ A A
3 nfv 1418 . . . . 5 x A A
4 eleq1 2097 . . . . 5 (x = A → (x AA A))
53, 4ceqsalg 2576 . . . 4 (A V → (x(x = Ax A) ↔ A A))
62, 5mtbiri 599 . . 3 (A V → ¬ x(x = Ax A))
7 elsn 3382 . . . . 5 (x {A} ↔ x = A)
8 olc 631 . . . . . 6 (x {A} → (x A x {A}))
9 elun 3078 . . . . . . 7 (x (A ∪ {A}) ↔ (x A x {A}))
10 ssid 2958 . . . . . . . . 9 AA
11 df-suc 4074 . . . . . . . . . . 11 suc A = (A ∪ {A})
1211eqeq1i 2044 . . . . . . . . . 10 (suc A = A ↔ (A ∪ {A}) = A)
13 sseq1 2960 . . . . . . . . . 10 ((A ∪ {A}) = A → ((A ∪ {A}) ⊆ AAA))
1412, 13sylbi 114 . . . . . . . . 9 (suc A = A → ((A ∪ {A}) ⊆ AAA))
1510, 14mpbiri 157 . . . . . . . 8 (suc A = A → (A ∪ {A}) ⊆ A)
1615sseld 2938 . . . . . . 7 (suc A = A → (x (A ∪ {A}) → x A))
179, 16syl5bir 142 . . . . . 6 (suc A = A → ((x A x {A}) → x A))
188, 17syl5 28 . . . . 5 (suc A = A → (x {A} → x A))
197, 18syl5bir 142 . . . 4 (suc A = A → (x = Ax A))
2019alrimiv 1751 . . 3 (suc A = Ax(x = Ax A))
216, 20nsyl3 556 . 2 (suc A = A → ¬ A V)
221, 21impbii 117 1 A V ↔ suc A = A)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 628  ∀wal 1240   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  {csn 3367  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-fal 1248  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-nul 3219  df-sn 3373  df-suc 4074 This theorem is referenced by: (None)
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