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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 6312 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: dfmpt 6316 fpar 7168 mapsnconst 7789 ixpsnf1o 7834 cda1dif 8881 infcda1 8898 s1co 13430 xpsc0 16043 xpsc1 16044 mat1f1o 20103 txdis 21245 pt1hmeo 21419 utop2nei 21864 utop3cls 21865 imasdsf1olem 21988 ex-xp 26685 poimirlem3 32582 poimirlem4 32583 poimirlem9 32588 poimirlem28 32607 grposnOLD 32851 dib0 35471 |
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