Theorem snsssn 4806

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 4792	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		sneqr.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	snnz 4742	. . . . 5 ⊢ {𝐴} ≠ ∅
4	3	neii 2960	. . . 4 ⊢ ¬ {𝐴} = ∅
5	4	pm2.21i 120	. . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵)
6	2	sneqr 4805	. . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)
7	5, 6	jaoi 870	. 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
8	1, 7	sylbi 220	1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∅c0 4286  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-ss 3922  df-nul 4287  df-sn 4590

This theorem is referenced by:  k0004lem3  44895