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Theorem bj-snglss 32658
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 32656 . . . . 5 (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦𝐴 𝑥 = {𝑦})
2 df-rex 2914 . . . . . 6 (∃𝑦𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦𝐴𝑥 = {𝑦}))
3 snssi 4315 . . . . . . . 8 (𝑦𝐴 → {𝑦} ⊆ 𝐴)
4 sseq1 3611 . . . . . . . . 9 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
54biimparc 504 . . . . . . . 8 (({𝑦} ⊆ 𝐴𝑥 = {𝑦}) → 𝑥𝐴)
63, 5sylan 488 . . . . . . 7 ((𝑦𝐴𝑥 = {𝑦}) → 𝑥𝐴)
76eximi 1759 . . . . . 6 (∃𝑦(𝑦𝐴𝑥 = {𝑦}) → ∃𝑦 𝑥𝐴)
82, 7sylbi 207 . . . . 5 (∃𝑦𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥𝐴)
91, 8sylbi 207 . . . 4 (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥𝐴)
10 ax5e 1838 . . . 4 (∃𝑦 𝑥𝐴𝑥𝐴)
119, 10syl 17 . . 3 (𝑥 ∈ sngl 𝐴𝑥𝐴)
12 selpw 4143 . . 3 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
1311, 12sylibr 224 . 2 (𝑥 ∈ sngl 𝐴𝑥 ∈ 𝒫 𝐴)
1413ssriv 3592 1 sngl 𝐴 ⊆ 𝒫 𝐴
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2909  wss 3560  𝒫 cpw 4136  {csn 4155  sngl bj-csngl 32653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-pw 4138  df-sn 4156  df-pr 4158  df-bj-sngl 32654
This theorem is referenced by:  bj-snglex  32661  bj-tagss  32668
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