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Mirrors > Home > ILE Home > Th. List > opexgOLD | GIF version |
Description: An ordered pair of sets is a set. This is a special case of opexg 3955 and new proofs should use opexg 3955 instead. (Contributed by Jim Kingdon, 19-Sep-2018.) (New usage is discouraged.) (Proof modification is discouraged.) TODO: replace its uses by uses of opexg 3955 and then remove it. |
Ref | Expression |
---|---|
opexgOLD | ⊢ ((A ∈ V ∧ B ∈ V) → 〈A, B〉 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3538 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → 〈A, B〉 = {{A}, {A, B}}) | |
2 | snexgOLD 3926 | . . . . 5 ⊢ (A ∈ V → {A} ∈ V) | |
3 | 2 | adantr 261 | . . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A} ∈ V) |
4 | prexgOLD 3937 | . . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V) | |
5 | 3, 4 | jca 290 | . . 3 ⊢ ((A ∈ V ∧ B ∈ V) → ({A} ∈ V ∧ {A, B} ∈ V)) |
6 | prexgOLD 3937 | . . 3 ⊢ (({A} ∈ V ∧ {A, B} ∈ V) → {{A}, {A, B}} ∈ V) | |
7 | 5, 6 | syl 14 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → {{A}, {A, B}} ∈ V) |
8 | 1, 7 | eqeltrd 2111 | 1 ⊢ ((A ∈ V ∧ B ∈ V) → 〈A, B〉 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 {csn 3367 {cpr 3368 〈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: otth2 3969 opeliunxp 4338 opbrop 4362 relsnop 4387 op2ndb 4747 opswapg 4750 elxp4 4751 elxp5 4752 fvsn 5301 resfunexg 5325 fliftel 5376 |
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