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Theorem bj-snglss 33266
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 33264 . . . . 5 (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦𝐴 𝑥 = {𝑦})
2 df-rex 3102 . . . . . 6 (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑥 = {𝑦}))
3 snssi 4529 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
4 sseq1 3823 . . . . . . . . 9 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
54biimparc 467 . . . . . . . 8 (({𝑦} ⊆ 𝐴𝑥 = {𝑦}) → 𝑥𝐴)
63, 5sylan 571 . . . . . . 7 ((𝑦𝐴𝑥 = {𝑦}) → 𝑥𝐴)
76eximi 1919 . . . . . 6 (∃𝑦(𝑦𝐴𝑥 = {𝑦}) → ∃𝑦 𝑥𝐴)
82, 7sylbi 208 . . . . 5 (∃𝑦𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥𝐴)
91, 8sylbi 208 . . . 4 (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥𝐴)
10 ax5e 2003 . . . 4 (∃𝑦 𝑥𝐴𝑥𝐴)
119, 10syl 17 . . 3 (𝑥 ∈ sngl 𝐴𝑥𝐴)
12 selpw 4358 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1311, 12sylibr 225 . 2 (𝑥 ∈ sngl 𝐴𝑥 ∈ 𝒫 𝐴)
1413ssriv 3802 1 sngl 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wex 1859  wcel 2156  wrex 3097  wss 3769  𝒫 cpw 4351  {csn 4370  sngl bj-csngl 33261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-pw 4353  df-sn 4371  df-pr 4373  df-bj-sngl 33262
This theorem is referenced by:  bj-snglex  33269  bj-tagss  33276
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