Theorem sspwuni 4050

Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)

Assertion

Ref	Expression
sspwuni	⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)

Proof of Theorem sspwuni

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2802	. . . 4 ⊢ 𝑥 ∈ V
2	1	elpw 3655	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
3	2	ralbii 2536	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
4		dfss3 3213	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵)
5		unissb 3918	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
6	3, 4, 5	3bitr4i 212	1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  𝒫 cpw 3649  ∪ cuni 3888

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-pw 3651  df-uni 3889

This theorem is referenced by:  pwssb  4051  elpwpw  4052  elpwuni  4055  rintm  4058  dftr4  4187  iotass  5299  tfrlemibfn  6485  tfr1onlembfn  6501  tfrcllembfn  6514  uniixp  6881  fipwssg  7162  unirnioo  10186  restid  13304  lssintclm  14369  topgele  14724  topontopn  14732  unitg  14757  epttop  14785  resttopon  14866  txuni2  14951  txdis  14972  unirnblps  15117  unirnbl  15118