Theorem bj-snglss 37144

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 37142	. . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦})
2		df-rex 3060	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}))
3		snssi 4763	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
4		sseq1 3958	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴))
5	4	biimparc 479	. . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
6	3, 5	sylan 581	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
7	6	eximi 1837	. . . . . 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 1914	. . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
11	9, 10	syl 17	. . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴)
12		velpw 4558	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	11, 12	sylibr 234	. 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴)
14	13	ssriv 3936	1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3059   ⊆ wss 3900  𝒫 cpw 4553  {csn 4579  sngl bj-csngl 37139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rex 3060  df-v 3441  df-un 3905  df-ss 3917  df-pw 4555  df-sn 4580  df-pr 4582  df-bj-sngl 37140

This theorem is referenced by:  bj-snglex  37147  bj-tagss  37154