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Theorem snsssn 4312
 Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1 𝐴 ∈ V
Assertion
Ref Expression
snsssn ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)

Proof of Theorem snsssn
StepHypRef Expression
1 sssn 4298 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 sneqr.1 . . . . . 6 𝐴 ∈ V
32snnz 4252 . . . . 5 {𝐴} ≠ ∅
43neii 2784 . . . 4 ¬ {𝐴} = ∅
54pm2.21i 115 . . 3 ({𝐴} = ∅ → 𝐴 = 𝐵)
62sneqr 4311 . . 3 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
75, 6jaoi 393 . 2 (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
81, 7sylbi 206 1 ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126 This theorem is referenced by:  k0004lem3  37467
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