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Mirrors > Home > MPE Home > Th. List > dfop | Structured version Visualization version 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.) (Avoid depending on this detail.) |
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 4338 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | mp2an 704 | 1 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 {cpr 4127 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-op 4132 |
This theorem is referenced by: opid 4359 elopOLD 4863 opi1 4864 opi2 4865 op1stb 4867 opeqsn 4892 opeqpr 4893 propssopi 4896 uniop 4902 xpsspw 5156 relop 5194 funopg 5836 |
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