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Mirrors > Home > ILE Home > Th. List > op2ndb | GIF version |
Description: Extract the second member of an ordered pair. Theorem 5.12(ii) of [Monk1] p. 52. (See op1stb 4175 to extract the first member and op2nda 4748 for an alternate 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 4746 | . . . . . 6 ⊢ ◡{〈A, B〉} = {〈B, A〉} |
4 | 3 | inteqi 3610 | . . . . 5 ⊢ ∩ ◡{〈A, B〉} = ∩ {〈B, A〉} |
5 | opexgOLD 3956 | . . . . . . 7 ⊢ ((B ∈ V ∧ A ∈ V) → 〈B, A〉 ∈ V) | |
6 | 2, 1, 5 | mp2an 402 | . . . . . 6 ⊢ 〈B, A〉 ∈ V |
7 | 6 | intsn 3641 | . . . . 5 ⊢ ∩ {〈B, A〉} = 〈B, A〉 |
8 | 4, 7 | eqtri 2057 | . . . 4 ⊢ ∩ ◡{〈A, B〉} = 〈B, A〉 |
9 | 8 | inteqi 3610 | . . 3 ⊢ ∩ ∩ ◡{〈A, B〉} = ∩ 〈B, A〉 |
10 | 9 | inteqi 3610 | . 2 ⊢ ∩ ∩ ∩ ◡{〈A, B〉} = ∩ ∩ 〈B, A〉 |
11 | 2, 1 | op1stb 4175 | . 2 ⊢ ∩ ∩ 〈B, A〉 = B |
12 | 10, 11 | eqtri 2057 | 1 ⊢ ∩ ∩ ∩ ◡{〈A, B〉} = B |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∈ wcel 1390 Vcvv 2551 {csn 3367 〈cop 3370 ∩ cint 3606 ◡ccnv 4287 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-int 3607 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 |
This theorem is referenced by: 2ndval2 5725 |
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