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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 4768 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2112 Vcvv 3398 {csn 4527 {cpr 4529 〈cop 4533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-v 3400 df-dif 3856 df-nul 4224 df-if 4426 df-op 4534 |
This theorem is referenced by: opi1 5337 opi2 5338 op1stb 5340 opeqpr 5373 propssopi 5376 uniop 5383 xpsspw 5664 relop 5704 funopg 6392 |
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