Theorem pwsnss 3907

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 3856	. . 3 ⊢ ((𝑥 = ∅ ∨ 𝑥 = {𝐴}) → 𝑥 ⊆ {𝐴})
2	1	ss2abi 3309	. 2 ⊢ {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})} ⊆ {𝑥 ∣ 𝑥 ⊆ {𝐴}}
3		dfpr2 3707	. 2 ⊢ {∅, {𝐴}} = {𝑥 ∣ (𝑥 = ∅ ∨ 𝑥 = {𝐴})}
4		df-pw 3670	. 2 ⊢ 𝒫 {𝐴} = {𝑥 ∣ 𝑥 ⊆ {𝐴}}
5	2, 3, 4	3sstr4i 3278	1 ⊢ {∅, {𝐴}} ⊆ 𝒫 {𝐴}

Syntax hints:   ∨ wo 716   = wceq 1398  {cab 2218   ⊆ wss 3210  ∅c0 3507  𝒫 cpw 3668  {csn 3688  {cpr 3689

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695

This theorem is referenced by:  pwpw0ss  3908