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| Mirrors > Home > MPE Home > Th. List > op2ndb | Structured version Visualization version GIF version | ||
| Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 5476 to extract the first member, op2nda 6248 for an alternate version, and op2nd 8023 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
| Ref | Expression |
|---|---|
| cnvsn.1 | ⊢ 𝐴 ∈ V |
| cnvsn.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| op2ndb | ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvsn.1 | . . . . . . 7 ⊢ 𝐴 ∈ V | |
| 2 | cnvsn.2 | . . . . . . 7 ⊢ 𝐵 ∈ V | |
| 3 | 1, 2 | cnvsn 6246 | . . . . . 6 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
| 4 | 3 | inteqi 4950 | . . . . 5 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = ∩ {〈𝐵, 𝐴〉} |
| 5 | opex 5469 | . . . . . 6 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
| 6 | 5 | intsn 4984 | . . . . 5 ⊢ ∩ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
| 7 | 4, 6 | eqtri 2765 | . . . 4 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
| 8 | 7 | inteqi 4950 | . . 3 ⊢ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ 〈𝐵, 𝐴〉 |
| 9 | 8 | inteqi 4950 | . 2 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ ∩ 〈𝐵, 𝐴〉 |
| 10 | 2, 1 | op1stb 5476 | . 2 ⊢ ∩ ∩ 〈𝐵, 𝐴〉 = 𝐵 |
| 11 | 9, 10 | eqtri 2765 | 1 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 ∩ cint 4946 ◡ccnv 5684 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-int 4947 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: 2ndval2 8032 |
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