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Theorem snsssn 4745
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 4732 . 2 ({𝐴} ⊆ {𝐵} ↔ ({𝐴} = ∅ ∨ {𝐴} = {𝐵}))
2 sneqr.1 . . . . . 6 𝐴 ∈ V
32snnz 4685 . . . . 5 {𝐴} ≠ ∅
43neii 3013 . . . 4 ¬ {𝐴} = ∅
54pm2.21i 119 . . 3 ({𝐴} = ∅ → 𝐴 = 𝐵)
62sneqr 4744 . . 3 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
75, 6jaoi 854 . 2 (({𝐴} = ∅ ∨ {𝐴} = {𝐵}) → 𝐴 = 𝐵)
81, 7sylbi 220 1 ({𝐴} ⊆ {𝐵} → 𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wcel 2114  Vcvv 3469  wss 3908  c0 4265  {csn 4539
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 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-nul 4266  df-sn 4540
This theorem is referenced by:  k0004lem3  40785
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