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Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5644. 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 4594 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4594 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5666 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5710 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 11158 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 12233 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3467 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 7092 | . . . . 5 ⊢ ({1} × {2}) = {⟨1, 2⟩} |
9 | 7nn 12252 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3467 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 7092 | . . . . 5 ⊢ ({1} × {7}) = {⟨1, 7⟩} |
12 | 8, 11 | uneq12i 4126 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({⟨1, 2⟩} ∪ {⟨1, 7⟩}) |
13 | df-pr 4594 | . . . 4 ⊢ {⟨1, 2⟩, ⟨1, 7⟩} = ({⟨1, 2⟩} ∪ {⟨1, 7⟩}) | |
14 | 12, 13 | eqtr4i 2768 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {⟨1, 2⟩, ⟨1, 7⟩} |
15 | 5nn 12246 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3467 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 7092 | . . . . 5 ⊢ ({5} × {2}) = {⟨5, 2⟩} |
18 | 16, 10 | xpsn 7092 | . . . . 5 ⊢ ({5} × {7}) = {⟨5, 7⟩} |
19 | 17, 18 | uneq12i 4126 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({⟨5, 2⟩} ∪ {⟨5, 7⟩}) |
20 | df-pr 4594 | . . . 4 ⊢ {⟨5, 2⟩, ⟨5, 7⟩} = ({⟨5, 2⟩} ∪ {⟨5, 7⟩}) | |
21 | 19, 20 | eqtr4i 2768 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {⟨5, 2⟩, ⟨5, 7⟩} |
22 | 14, 21 | uneq12i 4126 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) |
23 | 3, 4, 22 | 3eqtri 2769 | 1 ⊢ ({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3913 {csn 4591 {cpr 4593 ⟨cop 4597 × cxp 5636 1c1 11059 ℕcn 12160 2c2 12215 5c5 12218 7c7 12220 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 ax-1cn 11116 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 |
This theorem is referenced by: (None) |
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