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Mirrors > Home > ILE 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 3703 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵) | |
2 | unissb 3705 | . . . 4 ⊢ (∪ 𝐵 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) | |
3 | 2 | biimpri 132 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴) |
4 | 1, 3 | anim12i 332 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴)) |
5 | eqss 3054 | . 2 ⊢ (𝐴 = ∪ 𝐵 ↔ (𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴)) | |
6 | 4, 5 | sylibr 133 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴) → 𝐴 = ∪ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 ∀wral 2370 ⊆ wss 3013 ∪ cuni 3675 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-v 2635 df-in 3019 df-ss 3026 df-uni 3676 |
This theorem is referenced by: unimax 3709 hashinfuni 10316 hashennnuni 10318 |
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