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Mirrors > Home > ILE Home > Th. List > opex | GIF version |
Description: An ordered pair of sets is a set. (Contributed by Jim Kingdon, 24-Sep-2018.) (Revised by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
opex.1 | ⊢ 𝐴 ∈ V |
opex.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opex | ⊢ 〈𝐴, 𝐵〉 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opex.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opexg 4120 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 ∈ V) | |
4 | 1, 2, 3 | mp2an 422 | 1 ⊢ 〈𝐴, 𝐵〉 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 〈cop 3500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 |
This theorem is referenced by: otth2 4133 opabid 4149 elopab 4150 opabm 4172 elvvv 4572 relsnop 4615 xpiindim 4646 raliunxp 4650 rexiunxp 4651 intirr 4895 xpmlem 4929 dmsnm 4974 dmsnopg 4980 cnvcnvsn 4985 op2ndb 4992 cnviinm 5050 funopg 5127 fsn 5560 fvsn 5583 idref 5626 oprabid 5771 dfoprab2 5786 rnoprab 5822 fo1st 6023 fo2nd 6024 eloprabi 6062 xporderlem 6096 cnvoprab 6099 dmtpos 6121 rntpos 6122 tpostpos 6129 iinerm 6469 th3qlem2 6500 elixpsn 6597 ensn1 6658 mapsnen 6673 xpsnen 6683 xpcomco 6688 xpassen 6692 xpmapenlem 6711 phplem2 6715 ac6sfi 6760 djuss 6923 genipdm 7292 ioof 9722 fsumcnv 11174 txdis1cn 12374 |
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