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Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5563. 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 4572 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4572 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5585 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5627 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 10639 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 11713 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3515 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 6905 | . . . . 5 ⊢ ({1} × {2}) = {〈1, 2〉} |
9 | 7nn 11732 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3515 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 6905 | . . . . 5 ⊢ ({1} × {7}) = {〈1, 7〉} |
12 | 8, 11 | uneq12i 4139 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({〈1, 2〉} ∪ {〈1, 7〉}) |
13 | df-pr 4572 | . . . 4 ⊢ {〈1, 2〉, 〈1, 7〉} = ({〈1, 2〉} ∪ {〈1, 7〉}) | |
14 | 12, 13 | eqtr4i 2849 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {〈1, 2〉, 〈1, 7〉} |
15 | 5nn 11726 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3515 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 6905 | . . . . 5 ⊢ ({5} × {2}) = {〈5, 2〉} |
18 | 16, 10 | xpsn 6905 | . . . . 5 ⊢ ({5} × {7}) = {〈5, 7〉} |
19 | 17, 18 | uneq12i 4139 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({〈5, 2〉} ∪ {〈5, 7〉}) |
20 | df-pr 4572 | . . . 4 ⊢ {〈5, 2〉, 〈5, 7〉} = ({〈5, 2〉} ∪ {〈5, 7〉}) | |
21 | 19, 20 | eqtr4i 2849 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {〈5, 2〉, 〈5, 7〉} |
22 | 14, 21 | uneq12i 4139 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
23 | 3, 4, 22 | 3eqtri 2850 | 1 ⊢ ({1, 5} × {2, 7}) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3936 {csn 4569 {cpr 4571 〈cop 4575 × cxp 5555 1c1 10540 ℕcn 11640 2c2 11695 5c5 11698 7c7 11700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-1cn 10597 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 |
This theorem is referenced by: (None) |
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