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Theorem snsssn 3959
 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 3949 . 2
2 sneqr.1 . . . . . 6
32snnz 3914 . . . . 5
4 df-ne 2600 . . . . 5
53, 4mpbi 200 . . . 4
65pm2.21i 125 . . 3
72sneqr 3958 . . 3
86, 7jaoi 369 . 2
91, 8sylbi 188 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 358   wceq 1652   wcel 1725   wne 2598  cvv 2948   wss 3312  c0 3620  csn 3806 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-sn 3812
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