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Theorem bj-snglss 34401
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 34399 . . . . 5 (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦𝐴 𝑥 = {𝑦})
2 df-rex 3115 . . . . . 6 (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑥 = {𝑦}))
3 snssi 4704 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
4 sseq1 3943 . . . . . . . . 9 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
54biimparc 483 . . . . . . . 8 (({𝑦} ⊆ 𝐴𝑥 = {𝑦}) → 𝑥𝐴)
63, 5sylan 583 . . . . . . 7 ((𝑦𝐴𝑥 = {𝑦}) → 𝑥𝐴)
76eximi 1836 . . . . . 6 (∃𝑦(𝑦𝐴𝑥 = {𝑦}) → ∃𝑦 𝑥𝐴)
82, 7sylbi 220 . . . . 5 (∃𝑦𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥𝐴)
91, 8sylbi 220 . . . 4 (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥𝐴)
10 ax5e 1913 . . . 4 (∃𝑦 𝑥𝐴𝑥𝐴)
119, 10syl 17 . . 3 (𝑥 ∈ sngl 𝐴𝑥𝐴)
12 velpw 4505 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1311, 12sylibr 237 . 2 (𝑥 ∈ sngl 𝐴𝑥 ∈ 𝒫 𝐴)
1413ssriv 3922 1 sngl 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wex 1781  wcel 2112  wrex 3110  wss 3884  𝒫 cpw 4500  {csn 4528  sngl bj-csngl 34396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-pw 4502  df-sn 4529  df-pr 4531  df-bj-sngl 34397
This theorem is referenced by:  bj-snglex  34404  bj-tagss  34411
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