Theorem snsssn 3874

Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)

Hypothesis

Ref	Expression
sneqr.1	⊢ A ∈ V

Assertion

Ref	Expression
snsssn	⊢ ({A} ⊆ {B} → A = B)

Proof of Theorem snsssn

Step	Hyp	Ref	Expression
1		sssn 3865	. 2 ⊢ ({A} ⊆ {B} ↔ ({A} = ∅ ∨ {A} = {B}))
2		sneqr.1	. . . . . 6 ⊢ A ∈ V
3	2	snnz 3835	. . . . 5 ⊢ {A} ≠ ∅
4		df-ne 2519	. . . . 5 ⊢ ({A} ≠ ∅ ↔ ¬ {A} = ∅)
5	3, 4	mpbi 199	. . . 4 ⊢ ¬ {A} = ∅
6	5	pm2.21i 123	. . 3 ⊢ ({A} = ∅ → A = B)
7	2	sneqr 3873	. . 3 ⊢ ({A} = {B} → A = B)
8	6, 7	jaoi 368	. 2 ⊢ (({A} = ∅ ∨ {A} = {B}) → A = B)
9	1, 8	sylbi 187	1 ⊢ ({A} ⊆ {B} → A = B)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860   ⊆ wss 3258  ∅c0 3551  {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-nan 1288  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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742

This theorem is referenced by: (None)