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Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5673. 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 4624 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4624 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5695 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5740 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 11209 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 12284 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3486 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 7132 | . . . . 5 ⊢ ({1} × {2}) = {⟨1, 2⟩} |
9 | 7nn 12303 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3486 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 7132 | . . . . 5 ⊢ ({1} × {7}) = {⟨1, 7⟩} |
12 | 8, 11 | uneq12i 4154 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({⟨1, 2⟩} ∪ {⟨1, 7⟩}) |
13 | df-pr 4624 | . . . 4 ⊢ {⟨1, 2⟩, ⟨1, 7⟩} = ({⟨1, 2⟩} ∪ {⟨1, 7⟩}) | |
14 | 12, 13 | eqtr4i 2755 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {⟨1, 2⟩, ⟨1, 7⟩} |
15 | 5nn 12297 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3486 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 7132 | . . . . 5 ⊢ ({5} × {2}) = {⟨5, 2⟩} |
18 | 16, 10 | xpsn 7132 | . . . . 5 ⊢ ({5} × {7}) = {⟨5, 7⟩} |
19 | 17, 18 | uneq12i 4154 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({⟨5, 2⟩} ∪ {⟨5, 7⟩}) |
20 | df-pr 4624 | . . . 4 ⊢ {⟨5, 2⟩, ⟨5, 7⟩} = ({⟨5, 2⟩} ∪ {⟨5, 7⟩}) | |
21 | 19, 20 | eqtr4i 2755 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {⟨5, 2⟩, ⟨5, 7⟩} |
22 | 14, 21 | uneq12i 4154 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) |
23 | 3, 4, 22 | 3eqtri 2756 | 1 ⊢ ({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3939 {csn 4621 {cpr 4623 ⟨cop 4627 × cxp 5665 1c1 11108 ℕcn 12211 2c2 12266 5c5 12269 7c7 12271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 ax-1cn 11165 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 |
This theorem is referenced by: (None) |
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