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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 3955 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → 〈A, B〉 ∈ V) | |
4 | 1, 2, 3 | mp2an 402 | 1 ⊢ 〈A, B〉 ∈ V |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1390 Vcvv 2551 〈cop 3370 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 |
This theorem is referenced by: opabid 3985 elopab 3986 opabm 4008 elvvv 4346 xpiindim 4416 raliunxp 4420 rexiunxp 4421 intirr 4654 xpmlem 4687 dmsnm 4729 dmsnopg 4735 cnvcnvsn 4740 cnviinm 4802 funopg 4877 fsn 5278 idref 5339 oprabid 5480 dfoprab2 5494 rnoprab 5529 fo1st 5726 fo2nd 5727 eloprabi 5764 xporderlem 5793 dmtpos 5812 rntpos 5813 tpostpos 5820 iinerm 6114 th3qlem2 6145 ensn1 6212 xpsnen 6231 xpcomco 6236 xpassen 6240 genipdm 6499 ioof 8610 |
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