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Mirrors > Home > MPE Home > Th. List > xpprsng | Structured version Visualization version GIF version |
Description: The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.) |
Ref | Expression |
---|---|
xpprsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4564 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | xpeq1i 5576 | . 2 ⊢ ({𝐴, 𝐵} × {𝐶}) = (({𝐴} ∪ {𝐵}) × {𝐶}) |
3 | xpsng 6896 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐶}) = {〈𝐴, 𝐶〉}) | |
4 | 3 | 3adant2 1127 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐶}) = {〈𝐴, 𝐶〉}) |
5 | xpsng 6896 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐵} × {𝐶}) = {〈𝐵, 𝐶〉}) | |
6 | 5 | 3adant1 1126 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐵} × {𝐶}) = {〈𝐵, 𝐶〉}) |
7 | 4, 6 | uneq12d 4140 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (({𝐴} × {𝐶}) ∪ ({𝐵} × {𝐶})) = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐶〉})) |
8 | xpundir 5616 | . . 3 ⊢ (({𝐴} ∪ {𝐵}) × {𝐶}) = (({𝐴} × {𝐶}) ∪ ({𝐵} × {𝐶})) | |
9 | df-pr 4564 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐶〉}) | |
10 | 7, 8, 9 | 3eqtr4g 2881 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (({𝐴} ∪ {𝐵}) × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
11 | 2, 10 | syl5eq 2868 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴, 𝐵} × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∪ cun 3934 {csn 4561 {cpr 4563 〈cop 4567 × cxp 5548 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 |
This theorem is referenced by: linds2eq 30936 zlmodzxz0 44397 ehl2eudisval0 44705 |
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