Theorem snsssn 4841

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 4826	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		sneqr.1	. . . . . 6 ⊢ 𝐴 ∈ V
3	2	snnz 4776	. . . . 5 ⊢ {𝐴} ≠ ∅
4	3	neii 2942	. . . 4 ⊢ ¬ {𝐴} = ∅
5	4	pm2.21i 119	. . 3 ⊢ ({𝐴} = ∅ → 𝐴 = 𝐵)
6	2	sneqr 4840	. . 3 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)
7	5, 6	jaoi 858	. 2 ⊢ (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
8	1, 7	sylbi 217	1 ⊢ ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∨ wo 848   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-sn 4627

This theorem is referenced by:  k0004lem3  44162