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Mirrors > Home > ILE Home > Th. List > opexg | GIF version |
Description: An ordered pair of sets is a set. (Contributed by Jim Kingdon, 11-Jan-2019.) |
Ref | Expression |
---|---|
opexg | ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → 〈A, B〉 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3538 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → 〈A, B〉 = {{A}, {A, B}}) | |
2 | elex 2560 | . . . . 5 ⊢ (A ∈ 𝑉 → A ∈ V) | |
3 | snexg 3927 | . . . . 5 ⊢ (A ∈ V → {A} ∈ V) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (A ∈ 𝑉 → {A} ∈ V) |
5 | 4 | adantr 261 | . . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {A} ∈ V) |
6 | elex 2560 | . . . 4 ⊢ (B ∈ 𝑊 → B ∈ V) | |
7 | prexg 3938 | . . . 4 ⊢ ((A ∈ V ∧ B ∈ V) → {A, B} ∈ V) | |
8 | 2, 6, 7 | syl2an 273 | . . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {A, B} ∈ V) |
9 | prexg 3938 | . . 3 ⊢ (({A} ∈ V ∧ {A, B} ∈ V) → {{A}, {A, B}} ∈ V) | |
10 | 5, 8, 9 | syl2anc 391 | . 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {{A}, {A, B}} ∈ V) |
11 | 1, 10 | eqeltrd 2111 | 1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → 〈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: opex 3957 otexg 3958 fliftel1 5377 oprabid 5480 eloprabga 5533 op1st 5715 op2nd 5716 ot1stg 5721 ot2ndg 5722 ot3rdgg 5723 elxp6 5738 mpt2fvex 5771 algrflem 5792 mpt2xopoveq 5796 brtposg 5810 tfr0 5878 tfrlemisucaccv 5880 tfrlemibxssdm 5882 tfrlemibfn 5883 tfrlemi14d 5888 mulpipq2 6355 enq0breq 6419 axcnre 6765 frec2uzrdg 8876 frecuzrdg0 8881 frecuzrdgsuc 8882 |
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