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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 4804 | . 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 1536 ∈ wcel 2113 Vcvv 3497 {csn 4570 {cpr 4572 〈cop 4576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-dif 3942 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-op 4577 |
This theorem is referenced by: opi1 5363 opi2 5364 op1stb 5366 opeqpr 5398 propssopi 5401 uniop 5408 xpsspw 5685 relop 5724 funopg 6392 |
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