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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 3141 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) | |
2 | uneqri.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | |
3 | 1, 2 | bitri 182 | . 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ 𝐶) |
4 | 3 | eqriv 2085 | 1 ⊢ (𝐴 ∪ 𝐵) = 𝐶 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∨ wo 664 = wceq 1289 ∈ wcel 1438 ∪ cun 2997 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-v 2621 df-un 3003 |
This theorem is referenced by: unidm 3143 uncom 3144 unass 3157 undi 3247 unab 3266 un0 3316 |
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