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Mirrors > Home > MPE Home > Th. List > djueq12 | Structured version Visualization version GIF version |
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 5710 | . . . 4 ⊢ (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵)) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵)) |
3 | xpeq2 5710 | . . . 4 ⊢ (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷)) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷)) |
5 | 2, 4 | uneq12d 4179 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))) |
6 | df-dju 9939 | . 2 ⊢ (𝐴 ⊔ 𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶)) | |
7 | df-dju 9939 | . 2 ⊢ (𝐵 ⊔ 𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)) | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∪ cun 3961 ∅c0 4339 {csn 4631 × cxp 5687 1oc1o 8498 ⊔ cdju 9936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-opab 5211 df-xp 5695 df-dju 9939 |
This theorem is referenced by: djueq1 9943 djueq2 9944 isfin5 10337 alephadd 10615 |
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