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Mirrors > Home > ILE Home > Th. List > djueq1 | GIF version |
Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) |
Ref | Expression |
---|---|
djueq1 | ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2117 | . 2 ⊢ 𝐶 = 𝐶 | |
2 | djueq12 6892 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐶) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) | |
3 | 1, 2 | mpan2 421 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ⊔ cdju 6890 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-opab 3960 df-xp 4515 df-dju 6891 |
This theorem is referenced by: enumct 6968 ctssexmid 6992 unct 11881 subctctexmid 13123 sbthom 13148 |
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