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Mirrors > Home > ILE Home > Th. List > uneqri | GIF version |
Description: Inference from membership to union. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
uneqri.1 | ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) |
Ref | Expression |
---|---|
uneqri | ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elun 3212 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
2 | uneqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 183 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2134 | 1 ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 697 = wceq 1331 ∈ wcel 1480 ∪ cun 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 |
This theorem is referenced by: unidm 3214 uncom 3215 unass 3228 undi 3319 unab 3338 un0 3391 |
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