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| Description: Example for df-xp 5690. 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 4628 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
| 2 | df-pr 4628 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
| 3 | 1, 2 | xpeq12i 5712 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) | 
| 4 | xpun 5758 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
| 5 | 1ex 11258 | . . . . . 6 ⊢ 1 ∈ V | |
| 6 | 2nn 12340 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
| 7 | 6 | elexi 3502 | . . . . . 6 ⊢ 2 ∈ V | 
| 8 | 5, 7 | xpsn 7160 | . . . . 5 ⊢ ({1} × {2}) = {〈1, 2〉} | 
| 9 | 7nn 12359 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
| 10 | 9 | elexi 3502 | . . . . . 6 ⊢ 7 ∈ V | 
| 11 | 5, 10 | xpsn 7160 | . . . . 5 ⊢ ({1} × {7}) = {〈1, 7〉} | 
| 12 | 8, 11 | uneq12i 4165 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({〈1, 2〉} ∪ {〈1, 7〉}) | 
| 13 | df-pr 4628 | . . . 4 ⊢ {〈1, 2〉, 〈1, 7〉} = ({〈1, 2〉} ∪ {〈1, 7〉}) | |
| 14 | 12, 13 | eqtr4i 2767 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {〈1, 2〉, 〈1, 7〉} | 
| 15 | 5nn 12353 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
| 16 | 15 | elexi 3502 | . . . . . 6 ⊢ 5 ∈ V | 
| 17 | 16, 7 | xpsn 7160 | . . . . 5 ⊢ ({5} × {2}) = {〈5, 2〉} | 
| 18 | 16, 10 | xpsn 7160 | . . . . 5 ⊢ ({5} × {7}) = {〈5, 7〉} | 
| 19 | 17, 18 | uneq12i 4165 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({〈5, 2〉} ∪ {〈5, 7〉}) | 
| 20 | df-pr 4628 | . . . 4 ⊢ {〈5, 2〉, 〈5, 7〉} = ({〈5, 2〉} ∪ {〈5, 7〉}) | |
| 21 | 19, 20 | eqtr4i 2767 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {〈5, 2〉, 〈5, 7〉} | 
| 22 | 14, 21 | uneq12i 4165 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) | 
| 23 | 3, 4, 22 | 3eqtri 2768 | 1 ⊢ ({1, 5} × {2, 7}) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ cun 3948 {csn 4625 {cpr 4627 〈cop 4631 × cxp 5682 1c1 11157 ℕcn 12267 2c2 12322 5c5 12325 7c7 12327 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-1cn 11214 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 | 
| This theorem is referenced by: (None) | 
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