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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 4803 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 {cpr 4571 〈cop 4575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-op 4576 |
This theorem is referenced by: opi1 5362 opi2 5363 op1stb 5365 opeqpr 5397 propssopi 5400 uniop 5407 xpsspw 5684 relop 5723 funopg 6391 |
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