Theorem snsssng 32606

Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.) (Revised by Thierry Arnoux, 11-Apr-2024.)

Assertion

Ref	Expression
snsssng	⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)

Proof of Theorem snsssng

Step	Hyp	Ref	Expression
1		sssn 4760	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		snnzg 4709	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	2	neneqd 2941	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} = ∅)
4	3	pm2.21d 121	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵))
5		sneqrg 4773	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
6	4, 5	jaod 866	. . 3 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵))
7	6	imp 408	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵)
8	1, 7	sylan2b 601	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885  ∅c0 4264  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-v 3435  df-dif 3888  df-ss 3902  df-nul 4265  df-sn 4559

This theorem is referenced by: (None)