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Theorem snsssng 32542
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 ((𝐴𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)

Proof of Theorem snsssng
StepHypRef Expression
1 sssn 4831 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 snnzg 4779 . . . . . 6 (𝐴𝑉 → {𝐴} ≠ ∅)
32neneqd 2943 . . . . 5 (𝐴𝑉 → ¬ {𝐴} = ∅)
43pm2.21d 121 . . . 4 (𝐴𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵))
5 sneqrg 4844 . . . 4 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
64, 5jaod 859 . . 3 (𝐴𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵))
76imp 406 . 2 ((𝐴𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵)
81, 7sylan2b 594 1 ((𝐴𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847   = wceq 1537  wcel 2106  wss 3963  c0 4339  {csn 4631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-ss 3980  df-nul 4340  df-sn 4632
This theorem is referenced by: (None)
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