Theorem snsssn 4799

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 4785	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		sneqr.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	snnz 4737	. . . . 5 ⊢ {𝐴} ≠ ∅
4	3	neii 2961	. . . 4 ⊢ ¬ {𝐴} = ∅
5	4	pm2.21i 120	. . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵)
6	2	sneqr 4798	. . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)
7	5, 6	jaoi 871	. 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
8	1, 7	sylbi 220	1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2146  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-ss 3901  df-dif 4050  df-nul 4282  df-sn 4589

This theorem is used by:  k0004lem3  45292