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Theorem pw0 3739
Description: Compute the power set of the empty set. Theorem 89 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
pw0 𝒫 ∅ = {∅}

Proof of Theorem pw0
StepHypRef Expression
1 ss0b 3462 . . 3 (𝑥 ⊆ ∅ ↔ 𝑥 = ∅)
21abbii 2293 . 2 {𝑥𝑥 ⊆ ∅} = {𝑥𝑥 = ∅}
3 df-pw 3577 . 2 𝒫 ∅ = {𝑥𝑥 ⊆ ∅}
4 df-sn 3598 . 2 {∅} = {𝑥𝑥 = ∅}
52, 3, 43eqtr4i 2208 1 𝒫 ∅ = {∅}
Colors of variables: wff set class
Syntax hints:   = wceq 1353  {cab 2163  wss 3129  c0 3422  𝒫 cpw 3575  {csn 3592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598
This theorem is referenced by:  p0ex  4188  sn0topon  13519  sn0cld  13568
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