Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-snglss Structured version   Visualization version   GIF version

Theorem bj-snglss 34179
Description: The singletonization of a class is included in its powerclass. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-snglss sngl 𝐴 ⊆ 𝒫 𝐴

Proof of Theorem bj-snglss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-elsngl 34177 . . . . 5 (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦𝐴 𝑥 = {𝑦})
2 df-rex 3141 . . . . . 6 (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑥 = {𝑦}))
3 snssi 4733 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
4 sseq1 3989 . . . . . . . . 9 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
54biimparc 480 . . . . . . . 8 (({𝑦} ⊆ 𝐴𝑥 = {𝑦}) → 𝑥𝐴)
63, 5sylan 580 . . . . . . 7 ((𝑦𝐴𝑥 = {𝑦}) → 𝑥𝐴)
76eximi 1826 . . . . . 6 (∃𝑦(𝑦𝐴𝑥 = {𝑦}) → ∃𝑦 𝑥𝐴)
82, 7sylbi 218 . . . . 5 (∃𝑦𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥𝐴)
91, 8sylbi 218 . . . 4 (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥𝐴)
10 ax5e 1904 . . . 4 (∃𝑦 𝑥𝐴𝑥𝐴)
119, 10syl 17 . . 3 (𝑥 ∈ sngl 𝐴𝑥𝐴)
12 velpw 4543 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1311, 12sylibr 235 . 2 (𝑥 ∈ sngl 𝐴𝑥 ∈ 𝒫 𝐴)
1413ssriv 3968 1 sngl 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  wss 3933  𝒫 cpw 4535  {csn 4557  sngl bj-csngl 34174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rex 3141  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-pw 4537  df-sn 4558  df-pr 4560  df-bj-sngl 34175
This theorem is referenced by:  bj-snglex  34182  bj-tagss  34189
  Copyright terms: Public domain W3C validator