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Theorem snsssn 4140
 Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1
Assertion
Ref Expression
snsssn

Proof of Theorem snsssn
StepHypRef Expression
1 sssn 4130 . 2
2 sneqr.1 . . . . . 6
32snnz 4090 . . . . 5
43neii 2626 . . . 4
54pm2.21i 135 . . 3
62sneqr 4139 . . 3
75, 6jaoi 381 . 2
81, 7sylbi 199 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wo 370   wceq 1444   wcel 1887  cvv 3045   wss 3404  c0 3731  csn 3968 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-v 3047  df-dif 3407  df-in 3411  df-ss 3418  df-nul 3732  df-sn 3969 This theorem is referenced by: (None)
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