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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 4128 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
4 | 1, 2, 3 | mp2an 422 | . . 3 ⊢ {𝐴, 𝐵} ∈ V |
5 | 4 | prid2 3625 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
6 | 1, 2 | dfop 3699 | . 2 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
7 | 5, 6 | eleqtrri 2213 | 1 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1480 Vcvv 2681 {csn 3522 {cpr 3523 〈cop 3525 |
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 2119 ax-sep 4041 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 |
This theorem is referenced by: uniopel 4173 opeluu 4366 elvvuni 4598 |
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