Theorem snsssng 32494

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 4775	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		snnzg 4724	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	2	neneqd 2933	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} = ∅)
4	3	pm2.21d 121	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵))
5		sneqrg 4788	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
6	4, 5	jaod 859	. . 3 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵))
7	6	imp 406	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵)
8	1, 7	sylan2b 594	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∅c0 4280  {csn 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3900  df-ss 3914  df-nul 4281  df-sn 4574

This theorem is referenced by: (None)