Theorem sneqr 1856

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15.

Hypothesis

Ref	Expression
sneqr.1	⊢ A ∈ V

Assertion

Ref	Expression
sneqr	⊢ ({A} = {B} → A = B)

Proof of Theorem sneqr

Step	Hyp	Ref	Expression
1		sneqr.1	. . . 4 ⊢ A ∈ V
2	1	snid 1830	. . 3 ⊢ A ∈ {A}
3		eleq2 1150	. . 3 ⊢ ({A} = {B} → (A ∈ {A} ↔ A ∈ {B}))
4	2, 3	mpbii 168	. 2 ⊢ ({A} = {B} → A ∈ {B})
5	1	elsnc 1826	. 2 ⊢ (A ∈ {B} ↔ A = B)
6	4, 5	sylib 173	1 ⊢ ({A} = {B} → A = B)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808

This theorem is referenced by:  opth2 1909  opthwiener 1914  canth2 3381

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

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