Theorem sspwuni 4053

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 2803	. . . 4 ⊢ 𝑥 ∈ V
2	1	elpw 3656	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
3	2	ralbii 2536	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
4		dfss3 3214	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵)
5		unissb 3921	. 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 3198  𝒫 cpw 3650  ∪ cuni 3891

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 2802  df-in 3204  df-ss 3211  df-pw 3652  df-uni 3892

This theorem is referenced by:  pwssb  4054  elpwpw  4055  elpwuni  4058  rintm  4061  dftr4  4190  iotass  5302  tfrlemibfn  6489  tfr1onlembfn  6505  tfrcllembfn  6518  uniixp  6885  fipwssg  7172  unirnioo  10201  restid  13326  lssintclm  14391  topgele  14746  topontopn  14754  unitg  14779  epttop  14807  resttopon  14888  txuni2  14973  txdis  14994  unirnblps  15139  unirnbl  15140