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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 6903 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉}) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ ({𝐴} × {𝐵}) = {〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 〈cop 4575 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 |
This theorem is referenced by: dfmpt 6908 fpar 7813 mapsnconst 8458 ixpsnf1o 8504 dju1dif 9600 infdju1 9617 s1co 14197 mat1f1o 21089 txdis 22242 pt1hmeo 22416 utop2nei 22861 utop3cls 22862 imasdsf1olem 22985 ex-xp 28217 poimirlem3 34897 poimirlem4 34898 poimirlem9 34903 poimirlem28 34922 grposnOLD 35162 dib0 38302 |
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