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Mirrors > Home > MPE Home > Th. List > xpsn | Structured version Visualization version GIF version |
Description: The Cartesian product of two singletons is the singleton consisting in the associated ordered pair. (Contributed by NM, 4-Nov-2006.) |
Ref | Expression |
---|---|
xpsn.1 | ⊢ 𝐴 ∈ V |
xpsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
xpsn | ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | xpsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | xpsng 6671 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 682 | 1 ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 Vcvv 3398 {csn 4398 〈cop 4404 × cxp 5353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 |
This theorem is referenced by: dfmpt 6675 fpar 7562 mapsnconst 8189 ixpsnf1o 8234 cda1dif 9333 infcda1 9350 s1co 13984 xpsc0 16606 xpsc1 16607 mat1f1o 20689 txdis 21844 pt1hmeo 22018 utop2nei 22462 utop3cls 22463 imasdsf1olem 22586 ex-xp 27868 poimirlem3 34038 poimirlem4 34039 poimirlem9 34044 poimirlem28 34063 grposnOLD 34305 dib0 37318 |
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