Theorem sucprc 4458

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

Step	Hyp	Ref	Expression
1		df-suc 4417	. . 3 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2		snprc 3697	. . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		uneq2 3320	. . . 4 ⊢ ({𝐴} = ∅ → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
4	2, 3	sylbi 121	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐴 ∪ {𝐴}) = (𝐴 ∪ ∅))
5	1, 4	eqtrid 2249	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = (𝐴 ∪ ∅))
6		un0 3493	. 2 ⊢ (𝐴 ∪ ∅) = 𝐴
7	5, 6	eqtrdi 2253	1 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∪ cun 3163  ∅c0 3459  {csn 3632  suc csuc 4411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-un 3169  df-nul 3460  df-sn 3638  df-suc 4417

This theorem is referenced by:  sucprcreg  4596  sucon  4600