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Theorem snsssng 31327
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 4784 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 snnzg 4733 . . . . . 6 (𝐴𝑉 → {𝐴} ≠ ∅)
32neneqd 2946 . . . . 5 (𝐴𝑉 → ¬ {𝐴} = ∅)
43pm2.21d 121 . . . 4 (𝐴𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵))
5 sneqrg 4795 . . . 4 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
64, 5jaod 857 . . 3 (𝐴𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵))
76imp 407 . 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 396  wo 845   = wceq 1541  wcel 2106  wss 3908  c0 4280  {csn 4584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-v 3445  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4281  df-sn 4585
This theorem is referenced by: (None)
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