![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ex-xp | Structured version Visualization version GIF version |
Description: Example for df-xp 5684. 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 4632 | . . 3 ⊢ {1, 5} = ({1} ∪ {5}) | |
2 | df-pr 4632 | . . 3 ⊢ {2, 7} = ({2} ∪ {7}) | |
3 | 1, 2 | xpeq12i 5706 | . 2 ⊢ ({1, 5} × {2, 7}) = (({1} ∪ {5}) × ({2} ∪ {7})) |
4 | xpun 5751 | . 2 ⊢ (({1} ∪ {5}) × ({2} ∪ {7})) = ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) | |
5 | 1ex 11240 | . . . . . 6 ⊢ 1 ∈ V | |
6 | 2nn 12315 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
7 | 6 | elexi 3491 | . . . . . 6 ⊢ 2 ∈ V |
8 | 5, 7 | xpsn 7150 | . . . . 5 ⊢ ({1} × {2}) = {⟨1, 2⟩} |
9 | 7nn 12334 | . . . . . . 7 ⊢ 7 ∈ ℕ | |
10 | 9 | elexi 3491 | . . . . . 6 ⊢ 7 ∈ V |
11 | 5, 10 | xpsn 7150 | . . . . 5 ⊢ ({1} × {7}) = {⟨1, 7⟩} |
12 | 8, 11 | uneq12i 4160 | . . . 4 ⊢ (({1} × {2}) ∪ ({1} × {7})) = ({⟨1, 2⟩} ∪ {⟨1, 7⟩}) |
13 | df-pr 4632 | . . . 4 ⊢ {⟨1, 2⟩, ⟨1, 7⟩} = ({⟨1, 2⟩} ∪ {⟨1, 7⟩}) | |
14 | 12, 13 | eqtr4i 2759 | . . 3 ⊢ (({1} × {2}) ∪ ({1} × {7})) = {⟨1, 2⟩, ⟨1, 7⟩} |
15 | 5nn 12328 | . . . . . . 7 ⊢ 5 ∈ ℕ | |
16 | 15 | elexi 3491 | . . . . . 6 ⊢ 5 ∈ V |
17 | 16, 7 | xpsn 7150 | . . . . 5 ⊢ ({5} × {2}) = {⟨5, 2⟩} |
18 | 16, 10 | xpsn 7150 | . . . . 5 ⊢ ({5} × {7}) = {⟨5, 7⟩} |
19 | 17, 18 | uneq12i 4160 | . . . 4 ⊢ (({5} × {2}) ∪ ({5} × {7})) = ({⟨5, 2⟩} ∪ {⟨5, 7⟩}) |
20 | df-pr 4632 | . . . 4 ⊢ {⟨5, 2⟩, ⟨5, 7⟩} = ({⟨5, 2⟩} ∪ {⟨5, 7⟩}) | |
21 | 19, 20 | eqtr4i 2759 | . . 3 ⊢ (({5} × {2}) ∪ ({5} × {7})) = {⟨5, 2⟩, ⟨5, 7⟩} |
22 | 14, 21 | uneq12i 4160 | . 2 ⊢ ((({1} × {2}) ∪ ({1} × {7})) ∪ (({5} × {2}) ∪ ({5} × {7}))) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) |
23 | 3, 4, 22 | 3eqtri 2760 | 1 ⊢ ({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∪ cun 3945 {csn 4629 {cpr 4631 ⟨cop 4635 × cxp 5676 1c1 11139 ℕcn 12242 2c2 12297 5c5 12300 7c7 12302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 ax-1cn 11196 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |