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Theorem snsssng 30287
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 4722 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 snnzg 4673 . . . . . 6 (𝐴𝑉 → {𝐴} ≠ ∅)
32neneqd 2995 . . . . 5 (𝐴𝑉 → ¬ {𝐴} = ∅)
43pm2.21d 121 . . . 4 (𝐴𝑉 → ({𝐴} = ∅ → 𝐴 = 𝐵))
5 sneqrg 4733 . . . 4 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
64, 5jaod 856 . . 3 (𝐴𝑉 → (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵))
76imp 410 . 2 ((𝐴𝑉 ∧ ({𝐴} = ∅ ∨ {𝐴} = {𝐵})) → 𝐴 = 𝐵)
81, 7sylan2b 596 1 ((𝐴𝑉 ∧ {𝐴} ⊆ {𝐵}) → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844   = wceq 1538  wcel 2112  wss 3884  c0 4246  {csn 4528
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ne 2991  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247  df-sn 4529
This theorem is referenced by: (None)
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