Theorem snsssn 4866

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 4851	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		sneqr.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	snnz 4801	. . . . 5 ⊢ {𝐴} ≠ ∅
4	3	neii 2948	. . . 4 ⊢ ¬ {𝐴} = ∅
5	4	pm2.21i 119	. . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵)
6	2	sneqr 4865	. . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)
7	5, 6	jaoi 856	. 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
8	1, 7	sylbi 217	1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-v 3490  df-dif 3979  df-ss 3993  df-nul 4353  df-sn 4649

This theorem is referenced by:  k0004lem3  44111