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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 4560 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | xpeq1i 5574 | . 2 ⊢ ({𝐴, 𝐵} × {𝐶}) = (({𝐴} ∪ {𝐵}) × {𝐶}) |
3 | xpsng 6893 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐶}) = {〈𝐴, 𝐶〉}) | |
4 | 3 | 3adant2 1123 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐴} × {𝐶}) = {〈𝐴, 𝐶〉}) |
5 | xpsng 6893 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐵} × {𝐶}) = {〈𝐵, 𝐶〉}) | |
6 | 5 | 3adant1 1122 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → ({𝐵} × {𝐶}) = {〈𝐵, 𝐶〉}) |
7 | 4, 6 | uneq12d 4137 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (({𝐴} × {𝐶}) ∪ ({𝐵} × {𝐶})) = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐶〉})) |
8 | xpundir 5614 | . . 3 ⊢ (({𝐴} ∪ {𝐵}) × {𝐶}) = (({𝐴} × {𝐶}) ∪ ({𝐵} × {𝐶})) | |
9 | df-pr 4560 | . . 3 ⊢ {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉} = ({〈𝐴, 𝐶〉} ∪ {〈𝐵, 𝐶〉}) | |
10 | 7, 8, 9 | 3eqtr4g 2878 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑈) → (({𝐴} ∪ {𝐵}) × {𝐶}) = {〈𝐴, 𝐶〉, 〈𝐵, 𝐶〉}) |
11 | 2, 10 | syl5eq 2865 | 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 1079 = wceq 1528 ∈ wcel 2105 ∪ cun 3931 {csn 4557 {cpr 4559 〈cop 4563 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 |
This theorem is referenced by: linds2eq 30868 zlmodzxz0 44332 ehl2eudisval0 44640 |
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