Theorem pwsnss 3887

Description: The power set of a singleton. (Contributed by Jim Kingdon, 12-Aug-2018.)

Assertion

Ref	Expression
pwsnss	⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴}

Proof of Theorem pwsnss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sssnr 3836	. . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
2	1	ss2abi 3299	. 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}}
3		dfpr2 3688	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4		df-pw 3654	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
5	2, 3, 4	3sstr4i 3268	1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴}

Syntax hints:   ∨ wo 715   = wceq 1397  {cab 2217   ⊆ wss 3200  ∅c0 3494  𝒫 cpw 3652  {csn 3669  {cpr 3670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676

This theorem is referenced by:  pwpw0ss  3888