Theorem sucprcreg 3451

Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Regularity).

Assertion

Ref	Expression
sucprcreg	⊢ (¬ A ∈ V ↔ suc A = A)

Proof of Theorem sucprcreg

Step	Hyp	Ref	Expression
1		sucprc 2297	. 2 ⊢ (¬ A ∈ V → suc A = A)
2		eirr 3450	. . . 4 ⊢ ¬ A ∈ A
3		ax-17 925	. . . . 5 ⊢ (A ∈ A → ∀x A ∈ A)
4		eleq1 1149	. . . . 5 ⊢ (x = A → (x ∈ A ↔ A ∈ A))
5	3, 4	ceqsalg 1362	. . . 4 ⊢ (A ∈ V → (∀x(x = A → x ∈ A) ↔ A ∈ A))
6	2, 5	mtbiri 539	. . 3 ⊢ (A ∈ V → ¬ ∀x(x = A → x ∈ A))
7		ssid 1519	. . . . . . . . 9 ⊢ A ⊆ A
8		df-suc 2205	. . . . . . . . . . 11 ⊢ suc A = (A ∪ {A})
9	8	cleq1i 1108	. . . . . . . . . 10 ⊢ (suc A = A ↔ (A ∪ {A}) = A)
10		sseq1 1521	. . . . . . . . . 10 ⊢ ((A ∪ {A}) = A → ((A ∪ {A}) ⊆ A ↔ A ⊆ A))
11	9, 10	sylbi 174	. . . . . . . . 9 ⊢ (suc A = A → ((A ∪ {A}) ⊆ A ↔ A ⊆ A))
12	7, 11	mpbiri 169	. . . . . . . 8 ⊢ (suc A = A → (A ∪ {A}) ⊆ A)
13	12	sseld 1506	. . . . . . 7 ⊢ (suc A = A → (x ∈ (A ∪ {A}) → x ∈ A))
14		elun 1601	. . . . . . 7 ⊢ (x ∈ (A ∪ {A}) ↔ (x ∈ A ∨ x ∈ {A}))
15	13, 14	syl5ibr 182	. . . . . 6 ⊢ (suc A = A → ((x ∈ A ∨ x ∈ {A}) → x ∈ A))
16		olc 224	. . . . . 6 ⊢ (x ∈ {A} → (x ∈ A ∨ x ∈ {A}))
17	15, 16	syl5 22	. . . . 5 ⊢ (suc A = A → (x ∈ {A} → x ∈ A))
18		elsn 1820	. . . . 5 ⊢ (x ∈ {A} ↔ x = A)
19	17, 18	syl5ibr 182	. . . 4 ⊢ (suc A = A → (x = A → x ∈ A))
20	19	19.21aiv 943	. . 3 ⊢ (suc A = A → ∀x(x = A → x ∈ A))
21	6, 20	nsyl3 104	. 2 ⊢ (suc A = A → ¬ A ∈ V)
22	1, 21	impbi 139	1 ⊢ (¬ A ∈ V ↔ suc A = A)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∪ cun 1485   ⊆ wss 1487  {csn 1808  suc csuc 2201

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-reg 1078

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-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-suc 2205

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