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Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5561. 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 4570 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4570 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5583 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5625 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 10637 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 11711 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3513 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 6903 | . . . . 5 ⊢ ({1} × {2}) = {〈1, 2〉} |
9 | 7nn 11730 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3513 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 6903 | . . . . 5 ⊢ ({1} × {7}) = {〈1, 7〉} |
12 | 8, 11 | uneq12i 4137 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({〈1, 2〉} ∪ {〈1, 7〉}) |
13 | df-pr 4570 | . . . 4 ⊢ {〈1, 2〉, 〈1, 7〉} = ({〈1, 2〉} ∪ {〈1, 7〉}) | |
14 | 12, 13 | eqtr4i 2847 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {〈1, 2〉, 〈1, 7〉} |
15 | 5nn 11724 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3513 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 6903 | . . . . 5 ⊢ ({5} × {2}) = {〈5, 2〉} |
18 | 16, 10 | xpsn 6903 | . . . . 5 ⊢ ({5} × {7}) = {〈5, 7〉} |
19 | 17, 18 | uneq12i 4137 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({〈5, 2〉} ∪ {〈5, 7〉}) |
20 | df-pr 4570 | . . . 4 ⊢ {〈5, 2〉, 〈5, 7〉} = ({〈5, 2〉} ∪ {〈5, 7〉}) | |
21 | 19, 20 | eqtr4i 2847 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {〈5, 2〉, 〈5, 7〉} |
22 | 14, 21 | uneq12i 4137 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
23 | 3, 4, 22 | 3eqtri 2848 | 1 ⊢ ({1, 5} × {2, 7}) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3934 {csn 4567 {cpr 4569 〈cop 4573 × cxp 5553 1c1 10538 ℕcn 11638 2c2 11693 5c5 11696 7c7 11698 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-1cn 10595 |
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 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 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 |
This theorem is referenced by: (None) |
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