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Theorem bj-snglss 33800
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 33798 . . . . 5 (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦𝐴 𝑥 = {𝑦})
2 df-rex 3094 . . . . . 6 (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑥 = {𝑦}))
3 snssi 4616 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
4 sseq1 3884 . . . . . . . . 9 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
54biimparc 472 . . . . . . . 8 (({𝑦} ⊆ 𝐴𝑥 = {𝑦}) → 𝑥𝐴)
63, 5sylan 572 . . . . . . 7 ((𝑦𝐴𝑥 = {𝑦}) → 𝑥𝐴)
76eximi 1797 . . . . . 6 (∃𝑦(𝑦𝐴𝑥 = {𝑦}) → ∃𝑦 𝑥𝐴)
82, 7sylbi 209 . . . . 5 (∃𝑦𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥𝐴)
91, 8sylbi 209 . . . 4 (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥𝐴)
10 ax5e 1871 . . . 4 (∃𝑦 𝑥𝐴𝑥𝐴)
119, 10syl 17 . . 3 (𝑥 ∈ sngl 𝐴𝑥𝐴)
12 selpw 4430 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1311, 12sylibr 226 . 2 (𝑥 ∈ sngl 𝐴𝑥 ∈ 𝒫 𝐴)
1413ssriv 3864 1 sngl 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 387   = wceq 1507  wex 1742  wcel 2050  wrex 3089  wss 3831  𝒫 cpw 4423  {csn 4442  sngl bj-csngl 33795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-pw 4425  df-sn 4443  df-pr 4445  df-bj-sngl 33796
This theorem is referenced by:  bj-snglex  33803  bj-tagss  33810
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