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Mirrors > Home > NFE Home > Th. List > elpwuni | GIF version |
Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006.) |
Ref | Expression |
---|---|
elpwuni | ⊢ (B ∈ A → (A ⊆ ℘B ↔ ∪A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspwuni 4051 | . 2 ⊢ (A ⊆ ℘B ↔ ∪A ⊆ B) | |
2 | unissel 3920 | . . . 4 ⊢ ((∪A ⊆ B ∧ B ∈ A) → ∪A = B) | |
3 | 2 | expcom 424 | . . 3 ⊢ (B ∈ A → (∪A ⊆ B → ∪A = B)) |
4 | eqimss 3323 | . . 3 ⊢ (∪A = B → ∪A ⊆ B) | |
5 | 3, 4 | impbid1 194 | . 2 ⊢ (B ∈ A → (∪A ⊆ B ↔ ∪A = B)) |
6 | 1, 5 | syl5bb 248 | 1 ⊢ (B ∈ A → (A ⊆ ℘B ↔ ∪A = B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 ∈ wcel 1710 ⊆ 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: (None) |
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