Theorem bj-snglss 36971

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 36969	. . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦})
2		df-rex 3071	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}))
3		snssi 4808	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
4		sseq1 4009	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴))
5	4	biimparc 479	. . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
6	3, 5	sylan 580	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
7	6	eximi 1835	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → ∃𝑦 𝑥 ⊆ 𝐴)
8	2, 7	sylbi 217	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥 ⊆ 𝐴)
9	1, 8	sylbi 217	. . . 4 ⊢ (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥 ⊆ 𝐴)
10		ax5e 1912	. . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
11	9, 10	syl 17	. . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴)
12		velpw 4605	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	11, 12	sylibr 234	. 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴)
14	13	ssriv 3987	1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  𝒫 cpw 4600  {csn 4626  sngl bj-csngl 36966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-pr 4629  df-bj-sngl 36967

This theorem is referenced by:  bj-snglex  36974  bj-tagss  36981