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| Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.) | 
| Ref | Expression | 
|---|---|
| djueq12 | ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | xpeq2 5705 | . . . 4 ⊢ (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵)) | |
| 2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵)) | 
| 3 | xpeq2 5705 | . . . 4 ⊢ (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷)) | |
| 4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷)) | 
| 5 | 2, 4 | uneq12d 4168 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))) | 
| 6 | df-dju 9942 | . 2 ⊢ (𝐴 ⊔ 𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶)) | |
| 7 | df-dju 9942 | . 2 ⊢ (𝐵 ⊔ 𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)) | |
| 8 | 5, 6, 7 | 3eqtr4g 2801 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∪ cun 3948 ∅c0 4332 {csn 4625 × cxp 5682 1oc1o 8500 ⊔ cdju 9939 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 df-opab 5205 df-xp 5690 df-dju 9942 | 
| This theorem is referenced by: djueq1 9946 djueq2 9947 isfin5 10340 alephadd 10618 | 
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