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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 5464 to extract the first member, op2nda 6220 for an alternate version, and op2nd 7980 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 6218 | . . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩} |
4 | 3 | inteqi 4947 | . . . . 5 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ {⟨𝐵, 𝐴⟩} |
5 | opex 5457 | . . . . . 6 ⊢ ⟨𝐵, 𝐴⟩ ∈ V | |
6 | 5 | intsn 4983 | . . . . 5 ⊢ ∩ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩ |
7 | 4, 6 | eqtri 2754 | . . . 4 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩ |
8 | 7 | inteqi 4947 | . . 3 ⊢ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ⟨𝐵, 𝐴⟩ |
9 | 8 | inteqi 4947 | . 2 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ∩ ⟨𝐵, 𝐴⟩ |
10 | 2, 1 | op1stb 5464 | . 2 ⊢ ∩ ∩ ⟨𝐵, 𝐴⟩ = 𝐵 |
11 | 9, 10 | eqtri 2754 | 1 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3468 {csn 4623 ⟨cop 4629 ∩ cint 4943 ◡ccnv 5668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-int 4944 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-cnv 5677 |
This theorem is referenced by: 2ndval2 7989 |
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