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Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5044. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.) |
Ref | Expression |
---|---|
ex-xp | ⊢ ({1, 5} × {2, 7}) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4128 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4128 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5061 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5099 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 9914 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 11062 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3186 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 6313 | . . . . 5 ⊢ ({1} × {2}) = {〈1, 2〉} |
9 | 7nn 11067 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3186 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 6313 | . . . . 5 ⊢ ({1} × {7}) = {〈1, 7〉} |
12 | 8, 11 | uneq12i 3727 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({〈1, 2〉} ∪ {〈1, 7〉}) |
13 | df-pr 4128 | . . . 4 ⊢ {〈1, 2〉, 〈1, 7〉} = ({〈1, 2〉} ∪ {〈1, 7〉}) | |
14 | 12, 13 | eqtr4i 2635 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {〈1, 2〉, 〈1, 7〉} |
15 | 5nn 11065 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3186 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 6313 | . . . . 5 ⊢ ({5} × {2}) = {〈5, 2〉} |
18 | 16, 10 | xpsn 6313 | . . . . 5 ⊢ ({5} × {7}) = {〈5, 7〉} |
19 | 17, 18 | uneq12i 3727 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({〈5, 2〉} ∪ {〈5, 7〉}) |
20 | df-pr 4128 | . . . 4 ⊢ {〈5, 2〉, 〈5, 7〉} = ({〈5, 2〉} ∪ {〈5, 7〉}) | |
21 | 19, 20 | eqtr4i 2635 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {〈5, 2〉, 〈5, 7〉} |
22 | 14, 21 | uneq12i 3727 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
23 | 3, 4, 22 | 3eqtri 2636 | 1 ⊢ ({1, 5} × {2, 7}) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 {csn 4125 {cpr 4127 〈cop 4131 × cxp 5036 1c1 9816 ℕcn 10897 2c2 10947 5c5 10950 7c7 10952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-1cn 9873 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 |
This theorem is referenced by: (None) |
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