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Theorem snsssn 4370
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 4356 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 sneqr.1 . . . . . 6 𝐴 ∈ V
32snnz 4307 . . . . 5 {𝐴} ≠ ∅
43neii 2795 . . . 4 ¬ {𝐴} = ∅
54pm2.21i 116 . . 3 ({𝐴} = ∅ → 𝐴 = 𝐵)
62sneqr 4369 . . 3 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
75, 6jaoi 394 . 2 (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
81, 7sylbi 207 1 ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1482  wcel 1989  Vcvv 3198  wss 3572  c0 3913  {csn 4175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-v 3200  df-dif 3575  df-in 3579  df-ss 3586  df-nul 3914  df-sn 4176
This theorem is referenced by:  k0004lem3  38273
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