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Theorem snsssn 4198
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 4188 . 2
2 sneqr.1 . . . . . 6
32snnz 4148 . . . . 5
43neii 2656 . . . 4
54pm2.21i 131 . . 3
62sneqr 4197 . . 3
75, 6jaoi 379 . 2
81, 7sylbi 195 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  \/wo 368  =wceq 1395  e.wcel 1818   cvv 3109  C_wss 3475   c0 3784  {csn 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3478  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030
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