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Theorem sucprc 4367
 Description: A proper class is its own successor. (Contributed by NM, 3-Apr-1995.)
Assertion
Ref Expression
sucprc 𝐴 ∈ V → suc 𝐴 = 𝐴)

Proof of Theorem sucprc
StepHypRef Expression
1 df-suc 4326 . . 3 suc 𝐴 = (𝐴 ∪ {𝐴})
2 snprc 3620 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
3 uneq2 3251 . . . 4 ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
42, 3sylbi 120 . . 3 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
51, 4syl5eq 2199 . 2 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅))
6 un0 3423 . 2 (𝐴 ∪ ∅) = 𝐴
75, 6eqtrdi 2203 1 𝐴 ∈ V → suc 𝐴 = 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1332   ∈ wcel 2125  Vcvv 2709   ∪ cun 3096  ∅c0 3390  {csn 3556  suc csuc 4320 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-un 3102  df-nul 3391  df-sn 3562  df-suc 4326 This theorem is referenced by:  sucprcreg  4502  sucon  4506
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