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Mirrors > Home > ILE Home > Th. List > sspwuni | GIF version |
Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
sspwuni | ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2684 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elpw 3511 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
3 | 2 | ralbii 2439 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
4 | dfss3 3082 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵) | |
5 | unissb 3761 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | |
6 | 3, 4, 5 | 3bitr4i 211 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∈ wcel 1480 ∀wral 2414 ⊆ wss 3066 𝒫 cpw 3505 ∪ cuni 3731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-in 3072 df-ss 3079 df-pw 3507 df-uni 3732 |
This theorem is referenced by: pwssb 3893 elpwpw 3894 elpwuni 3897 rintm 3900 dftr4 4026 iotass 5100 tfrlemibfn 6218 tfr1onlembfn 6234 tfrcllembfn 6247 uniixp 6608 fipwssg 6860 unirnioo 9749 restid 12120 topgele 12185 topontopn 12193 unitg 12220 epttop 12248 resttopon 12329 txuni2 12414 txdis 12435 unirnblps 12580 unirnbl 12581 |
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