New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > sspwuni | GIF version |
Description: Subclass relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
sspwuni | ⊢ (A ⊆ ℘B ↔ ∪A ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2862 | . . . 4 ⊢ x ∈ V | |
2 | 1 | elpw 3728 | . . 3 ⊢ (x ∈ ℘B ↔ x ⊆ B) |
3 | 2 | ralbii 2638 | . 2 ⊢ (∀x ∈ A x ∈ ℘B ↔ ∀x ∈ A x ⊆ B) |
4 | dfss3 3263 | . 2 ⊢ (A ⊆ ℘B ↔ ∀x ∈ A x ∈ ℘B) | |
5 | unissb 3921 | . 2 ⊢ (∪A ⊆ B ↔ ∀x ∈ A x ⊆ B) | |
6 | 3, 4, 5 | 3bitr4i 268 | 1 ⊢ (A ⊆ ℘B ↔ ∪A ⊆ B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∈ wcel 1710 ∀wral 2614 ⊆ wss 3257 ℘cpw 3722 ∪cuni 3891 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-pw 3724 df-uni 3892 |
This theorem is referenced by: pwssb 4052 elpwuni 4053 rintn0 4056 qsss 5986 |
Copyright terms: Public domain | W3C validator |