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Mirrors > Home > NFE Home > Th. List > ssunieq | GIF version |
Description: Relationship implying union. (Contributed by NM, 10-Nov-1999.) |
Ref | Expression |
---|---|
ssunieq | ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → A = ∪B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 3919 | . . 3 ⊢ (A ∈ B → A ⊆ ∪B) | |
2 | unissb 3921 | . . . 4 ⊢ (∪B ⊆ A ↔ ∀x ∈ B x ⊆ A) | |
3 | 2 | biimpri 197 | . . 3 ⊢ (∀x ∈ B x ⊆ A → ∪B ⊆ A) |
4 | 1, 3 | anim12i 549 | . 2 ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → (A ⊆ ∪B ∧ ∪B ⊆ A)) |
5 | eqss 3287 | . 2 ⊢ (A = ∪B ↔ (A ⊆ ∪B ∧ ∪B ⊆ A)) | |
6 | 4, 5 | sylibr 203 | 1 ⊢ ((A ∈ B ∧ ∀x ∈ B x ⊆ A) → A = ∪B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ∀wral 2614 ⊆ wss 3257 ∪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-uni 3892 |
This theorem is referenced by: unimax 3925 |
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