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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 4150 | . 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 1480 Vcvv 2686 〈cop 3530 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: otth2 4163 opabid 4179 elopab 4180 opabm 4202 elvvv 4602 relsnop 4645 xpiindim 4676 raliunxp 4680 rexiunxp 4681 intirr 4925 xpmlem 4959 dmsnm 5004 dmsnopg 5010 cnvcnvsn 5015 op2ndb 5022 cnviinm 5080 funopg 5157 fsn 5592 fvsn 5615 idref 5658 oprabid 5803 dfoprab2 5818 rnoprab 5854 fo1st 6055 fo2nd 6056 eloprabi 6094 xporderlem 6128 cnvoprab 6131 dmtpos 6153 rntpos 6154 tpostpos 6161 iinerm 6501 th3qlem2 6532 elixpsn 6629 ensn1 6690 mapsnen 6705 xpsnen 6715 xpcomco 6720 xpassen 6724 xpmapenlem 6743 phplem2 6747 ac6sfi 6792 djuss 6955 genipdm 7324 ioof 9754 fsumcnv 11206 txdis1cn 12447 |
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