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Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5120. 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 4180 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4180 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5137 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5176 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 10035 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 11185 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3213 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 6407 | . . . . 5 ⊢ ({1} × {2}) = {〈1, 2〉} |
9 | 7nn 11190 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3213 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 6407 | . . . . 5 ⊢ ({1} × {7}) = {〈1, 7〉} |
12 | 8, 11 | uneq12i 3765 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({〈1, 2〉} ∪ {〈1, 7〉}) |
13 | df-pr 4180 | . . . 4 ⊢ {〈1, 2〉, 〈1, 7〉} = ({〈1, 2〉} ∪ {〈1, 7〉}) | |
14 | 12, 13 | eqtr4i 2647 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {〈1, 2〉, 〈1, 7〉} |
15 | 5nn 11188 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3213 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 6407 | . . . . 5 ⊢ ({5} × {2}) = {〈5, 2〉} |
18 | 16, 10 | xpsn 6407 | . . . . 5 ⊢ ({5} × {7}) = {〈5, 7〉} |
19 | 17, 18 | uneq12i 3765 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({〈5, 2〉} ∪ {〈5, 7〉}) |
20 | df-pr 4180 | . . . 4 ⊢ {〈5, 2〉, 〈5, 7〉} = ({〈5, 2〉} ∪ {〈5, 7〉}) | |
21 | 19, 20 | eqtr4i 2647 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {〈5, 2〉, 〈5, 7〉} |
22 | 14, 21 | uneq12i 3765 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
23 | 3, 4, 22 | 3eqtri 2648 | 1 ⊢ ({1, 5} × {2, 7}) = ({〈1, 2〉, 〈1, 7〉} ∪ {〈5, 2〉, 〈5, 7〉}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∪ cun 3572 {csn 4177 {cpr 4179 〈cop 4183 × cxp 5112 1c1 9937 ℕcn 11020 2c2 11070 5c5 11073 7c7 11075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 |
This theorem is referenced by: (None) |
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