Theorem snsssng 30259

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 4733	. 2 ⊢ ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2		snnzg 4684	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≠ ∅)
3	2	neneqd 3011	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ¬ {𝐴} = ∅)
4	3	pm2.21d 121	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵))
5		sneqrg 4744	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
6	4, 5	jaod 855	. . 3 ⊢ (𝐴 ∈ 𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵))
7	6	imp 409	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵)
8	1, 7	sylan2b 595	1 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ⊆ wss 3912  ∅c0 4267  {csn 4541

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-v 3475  df-dif 3915  df-in 3919  df-ss 3928  df-nul 4268  df-sn 4542

This theorem is referenced by: (None)