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Mirrors > Home > MPE Home > Th. List > xpsn | Structured version Visualization version GIF version |
Description: The Cartesian product of two singletons. (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 6552 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 672 | 1 ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 ∈ wcel 2145 Vcvv 3351 {csn 4317 〈cop 4323 × cxp 5248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 |
This theorem is referenced by: dfmpt 6556 fpar 7436 mapsnconst 8061 ixpsnf1o 8106 cda1dif 9204 infcda1 9221 s1co 13788 xpsc0 16428 xpsc1 16429 mat1f1o 20502 txdis 21656 pt1hmeo 21830 utop2nei 22274 utop3cls 22275 imasdsf1olem 22398 ex-xp 27635 poimirlem3 33744 poimirlem4 33745 poimirlem9 33750 poimirlem28 33769 grposnOLD 34011 dib0 36972 |
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