Theorem bj-snglss 36307

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 36305	. . . . 5 ⊢ (𝑥 ∈ sngl 𝐴 ↔ ∃𝑦 ∈ 𝐴 𝑥 = {𝑦})
2		df-rex 3063	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}))
3		snssi 4803	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → {𝑦} ⊆ 𝐴)
4		sseq1 3999	. . . . . . . . 9 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴))
5	4	biimparc 479	. . . . . . . 8 ⊢ (({𝑦} ⊆ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
6	3, 5	sylan 579	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → 𝑥 ⊆ 𝐴)
7	6	eximi 1829	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑥 = {𝑦}) → ∃𝑦 𝑥 ⊆ 𝐴)
8	2, 7	sylbi 216	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 = {𝑦} → ∃𝑦 𝑥 ⊆ 𝐴)
9	1, 8	sylbi 216	. . . 4 ⊢ (𝑥 ∈ sngl 𝐴 → ∃𝑦 𝑥 ⊆ 𝐴)
10		ax5e 1907	. . . 4 ⊢ (∃𝑦 𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐴)
11	9, 10	syl 17	. . 3 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ⊆ 𝐴)
12		velpw 4599	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
13	11, 12	sylibr 233	. 2 ⊢ (𝑥 ∈ sngl 𝐴 → 𝑥 ∈ 𝒫 𝐴)
14	13	ssriv 3978	1 ⊢ sngl 𝐴 ⊆ 𝒫 𝐴

Syntax hints:   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃wrex 3062   ⊆ wss 3940  𝒫 cpw 4594  {csn 4620  sngl bj-csngl 36302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-v 3468  df-un 3945  df-in 3947  df-ss 3957  df-pw 4596  df-sn 4621  df-pr 4623  df-bj-sngl 36303

This theorem is referenced by:  bj-snglex  36310  bj-tagss  36317