Theorem pwidg 3666

Description: Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion

Ref	Expression
pwidg	⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)

Proof of Theorem pwidg

Step	Hyp	Ref	Expression
1		ssid 3247	. 2 ⊢ 𝐴 ⊆ 𝐴
2		elpwg 3660	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐴 ↔ 𝐴 ⊆ 𝐴))
3	1, 2	mpbiri 168	1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2202   ⊆ wss 3200  𝒫 cpw 3652

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-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-in 3206  df-ss 3213  df-pw 3654

This theorem is referenced by:  pwid  3667  axpweq  4261  baspartn  14780  epttop  14820  isopn3  14855