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

Theorem sucprcreg 4467
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 4337 . 2 𝐴 ∈ V → suc 𝐴 = 𝐴)
2 elirr 4459 . . . 4 ¬ 𝐴𝐴
3 nfv 1508 . . . . 5 𝑥 𝐴𝐴
4 eleq1 2202 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐴𝐴𝐴))
53, 4ceqsalg 2714 . . . 4 (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴𝑥𝐴) ↔ 𝐴𝐴))
62, 5mtbiri 664 . . 3 (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴𝑥𝐴))
7 velsn 3544 . . . . 5 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8 olc 700 . . . . . 6 (𝑥 ∈ {𝐴} → (𝑥𝐴𝑥 ∈ {𝐴}))
9 elun 3217 . . . . . . 7 (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥𝐴𝑥 ∈ {𝐴}))
10 ssid 3117 . . . . . . . . 9 𝐴𝐴
11 df-suc 4296 . . . . . . . . . . 11 suc 𝐴 = (𝐴 ∪ {𝐴})
1211eqeq1i 2147 . . . . . . . . . 10 (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13 sseq1 3120 . . . . . . . . . 10 ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1412, 13sylbi 120 . . . . . . . . 9 (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴𝐴𝐴))
1510, 14mpbiri 167 . . . . . . . 8 (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
1615sseld 3096 . . . . . . 7 (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥𝐴))
179, 16syl5bir 152 . . . . . 6 (suc 𝐴 = 𝐴 → ((𝑥𝐴𝑥 ∈ {𝐴}) → 𝑥𝐴))
188, 17syl5 32 . . . . 5 (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥𝐴))
197, 18syl5bir 152 . . . 4 (suc 𝐴 = 𝐴 → (𝑥 = 𝐴𝑥𝐴))
2019alrimiv 1846 . . 3 (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴𝑥𝐴))
216, 20nsyl3 615 . 2 (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
221, 21impbii 125 1 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 697  wal 1329   = wceq 1331  wcel 1480  Vcvv 2686  cun 3069  wss 3071  {csn 3527  suc csuc 4290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-setind 4455
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-suc 4296
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator