Theorem sucid 2304

Description: A set belongs to its successor.

Hypothesis

Ref	Expression
sucid.1	⊢ A ∈ V

Assertion

Ref	Expression
sucid	⊢ A ∈ suc A

Proof of Theorem sucid

Step	Hyp	Ref	Expression
1		sucid.1	. . 3 ⊢ A ∈ V
2	1	snid 1830	. 2 ⊢ A ∈ {A}
3		df-suc 2205	. . . . . 6 ⊢ suc A = (A ∪ {A})
4	3	eleq2i 1153	. . . . 5 ⊢ (A ∈ suc A ↔ A ∈ (A ∪ {A}))
5		elun 1601	. . . . 5 ⊢ (A ∈ (A ∪ {A}) ↔ (A ∈ A ∨ A ∈ {A}))
6	4, 5	bitr 151	. . . 4 ⊢ (A ∈ suc A ↔ (A ∈ A ∨ A ∈ {A}))
7	6	biimpr 134	. . 3 ⊢ ((A ∈ A ∨ A ∈ {A}) → A ∈ suc A)
8	7	olci 227	. 2 ⊢ (A ∈ {A} → A ∈ suc A)
9	2, 8	ax-mp 6	1 ⊢ A ∈ suc A

Syntax hints:   ∨ wo 195   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485  {csn 1808  suc csuc 2201

This theorem is referenced by:  sucidg 2305  eqelsuc 2307  unon 2338  onuninsuc 2356  nlimsuc 2363  peano5 2394  tfinds 2401  tz7.44-2 2967  oawordeulem 3156  oalimcl 3162  phplem5 3407  php 3409  fiint 3445  inf0 3457  r1val1 3502  rankwflem 3509  rankr1 3518  cardlim 3657  cardaleph 3690  1lt2pi 3826  indpi 3828

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-suc 2205

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