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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 4141 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
4 | 1, 2, 3 | mp2an 423 | . . 3 ⊢ {𝐴, 𝐵} ∈ V |
5 | 4 | prid2 3638 | . 2 ⊢ {𝐴, 𝐵} ∈ {{𝐴}, {𝐴, 𝐵}} |
6 | 1, 2 | dfop 3712 | . 2 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
7 | 5, 6 | eleqtrri 2216 | 1 ⊢ {𝐴, 𝐵} ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 Vcvv 2689 {csn 3532 {cpr 3533 〈cop 3535 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 |
This theorem is referenced by: uniopel 4186 opeluu 4379 elvvuni 4611 |
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