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Theorem snsspw 3877
 Description: The singleton of a class is a subset of its power class. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
snsspw {A} A

Proof of Theorem snsspw
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eqimss 3323 . . 3 (x = Ax A)
2 elsn 3748 . . 3 (x {A} ↔ x = A)
3 df-pw 3724 . . . 4 A = {x x A}
43abeq2i 2460 . . 3 (x Ax A)
51, 2, 43imtr4i 257 . 2 (x {A} → x A)
65ssriv 3277 1 {A} A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-sn 3741 This theorem is referenced by: (None)
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