Theorem pwpwssunieq 4082

Description: The class of sets whose union is equal to a given class is included in the double power class of that class. (Contributed by BJ, 29-Apr-2021.)

Assertion

Ref	Expression
pwpwssunieq	⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem pwpwssunieq

Step	Hyp	Ref	Expression
1		eqimss 3294	. . 3 ⊢ (∪ 𝑥 = 𝐴 → ∪ 𝑥 ⊆ 𝐴)
2	1	ss2abi 3312	. 2 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
3		pwpwab 4081	. 2 ⊢ 𝒫 𝒫 𝐴 = {𝑥 ∣ ∪ 𝑥 ⊆ 𝐴}
4	2, 3	sseqtrri 3275	1 ⊢ {𝑥 ∣ ∪ 𝑥 = 𝐴} ⊆ 𝒫 𝒫 𝐴

Syntax hints:   = wceq 1398  {cab 2220   ⊆ wss 3213  𝒫 cpw 3671  ∪ cuni 3916

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 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 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-in 3219  df-ss 3226  df-pw 3673  df-uni 3917

This theorem is referenced by:  toponsspwpwg  14936  dmtopon  14937