Theorem snsssn 4808

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 4794	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		sneqr.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	snnz 4744	. . . . 5 ⊢ {𝐴} ≠ ∅
4	3	neii 2962	. . . 4 ⊢ ¬ {𝐴} = ∅
5	4	pm2.21i 120	. . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵)
6	2	sneqr 4807	. . 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 1570   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  ∅c0 4286  {csn 4591

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-v 3459  df-dif 3909  df-ss 3923  df-nul 4287  df-sn 4592

This theorem is used by:  k0004lem3  44935