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Mirrors > Home > ILE Home > Th. List > opi2 | GIF version |
Description: One of the two elements of an ordered pair. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opi1.1 | ⊢ 𝐴 ∈ V |
opi1.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
opi2 | ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opi1.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | opi1.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | prexg 4103 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
4 | 1, 2, 3 | mp2an 422 | . . 3 ⊢ {𝐴, 𝐵} ∈ V |
5 | 4 | prid2 3600 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
6 | 1, 2 | dfop 3674 | . 2 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
7 | 5, 6 | eleqtrri 2193 | 1 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 Vcvv 2660 {csn 3497 {cpr 3498 〈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-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-sn 3503 df-pr 3504 df-op 3506 |
This theorem is referenced by: uniopel 4148 opeluu 4341 elvvuni 4573 |
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