Theorem sneqr 3873

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)

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 3761	. . 3 ⊢ A ∈ {A}
3		eleq2 2414	. . 3 ⊢ ({A} = {B} → (A ∈ {A} ↔ A ∈ {B}))
4	2, 3	mpbii 202	. 2 ⊢ ({A} = {B} → A ∈ {B})
5	1	elsnc 3757	. 2 ⊢ (A ∈ {B} ↔ A = B)
6	4, 5	sylib 188	1 ⊢ ({A} = {B} → A = B)

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sn 3742

This theorem is referenced by:  snsssn  3874  sneqrg  3875