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Mirrors > Home > ILE Home > Th. List > dfop | GIF version |
Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.) |
Ref | Expression |
---|---|
dfop.1 | ⊢ 𝐴 ∈ V |
dfop.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
dfop | ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfop.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | dfop.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | dfopg 3711 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | mp2an 423 | 1 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ 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-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
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-v 2691 df-op 3541 |
This theorem is referenced by: opid 3731 elop 4161 opi1 4162 opi2 4163 opeqsn 4182 opeqpr 4183 uniop 4185 op1stb 4407 xpsspw 4659 relop 4697 funopg 5165 |
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