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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 2755 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elpw 3599 | . . 3 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) |
3 | 2 | ralbii 2496 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) |
4 | dfss3 3160 | . 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝒫 𝐵) | |
5 | unissb 3857 | . 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵) | |
6 | 3, 4, 5 | 3bitr4i 212 | 1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∈ wcel 2160 ∀wral 2468 ⊆ wss 3144 𝒫 cpw 3593 ∪ cuni 3827 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-v 2754 df-in 3150 df-ss 3157 df-pw 3595 df-uni 3828 |
This theorem is referenced by: pwssb 3990 elpwpw 3991 elpwuni 3994 rintm 3997 dftr4 4124 iotass 5216 tfrlemibfn 6357 tfr1onlembfn 6373 tfrcllembfn 6386 uniixp 6751 fipwssg 7012 unirnioo 10009 restid 12766 lssintclm 13725 topgele 14014 topontopn 14022 unitg 14047 epttop 14075 resttopon 14156 txuni2 14241 txdis 14262 unirnblps 14407 unirnbl 14408 |
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