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Theorem sspwuni 3739
 Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.)
Assertion
Ref Expression
sspwuni (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)

Proof of Theorem sspwuni
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . 4 𝑥 ∈ V
21elpw 3365 . . 3 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
32ralbii 2330 . 2 (∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
4 dfss3 2935 . 2 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ 𝒫 𝐵)
5 unissb 3610 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
63, 4, 53bitr4i 201 1 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  ∀wral 2306   ⊆ wss 2917  𝒫 cpw 3359  ∪ cuni 3580 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-uni 3581 This theorem is referenced by:  pwssb  3740  elpwuni  3741  rintm  3744  dftr4  3859  iotass  4884  tfrlemibfn  5942  unirnioo  8842
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