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.

Step	Hyp	Ref	Expression
1		bj-elsngl 33798	. . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦})
2		df-rex 3094	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}))
3		snssi 4616	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
4		sseq1 3884	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴))
5	4	biimparc 472	. . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
6	3, 5	sylan 572	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
7	6	eximi 1797	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → ∃𝑦 𝑥 ⊆ 𝐴)
8	2, 7	sylbi 209	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥 ⊆ 𝐴)
9	1, 8	sylbi 209	. . . 4 ⊢ (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥 ⊆ 𝐴)
10		ax5e 1871	. . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
11	9, 10	syl 17	. . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴)
12		selpw 4430	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	11, 12	sylibr 226	. 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴)
14	13	ssriv 3864	1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴

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