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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 6878 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 691 | 1 ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 |
This theorem is referenced by: dfmpt 6883 fpar 7794 mapsnconst 8439 ixpsnf1o 8485 dju1dif 9583 infdju1 9600 s1co 14186 mat1f1o 21083 txdis 22237 pt1hmeo 22411 utop2nei 22856 utop3cls 22857 imasdsf1olem 22980 ex-xp 28221 poimirlem3 35060 poimirlem4 35061 poimirlem9 35066 poimirlem28 35085 grposnOLD 35320 dib0 38460 |
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