Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4394 to extract the first member and op2nda 5018 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 5016 | . . . . . 6 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
4 | 3 | inteqi 3770 | . . . . 5 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = ∩ {〈𝐵, 𝐴〉} |
5 | 2, 1 | opex 4146 | . . . . . 6 ⊢ 〈𝐵, 𝐴〉 ∈ V |
6 | 5 | intsn 3801 | . . . . 5 ⊢ ∩ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
7 | 4, 6 | eqtri 2158 | . . . 4 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
8 | 7 | inteqi 3770 | . . 3 ⊢ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ 〈𝐵, 𝐴〉 |
9 | 8 | inteqi 3770 | . 2 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ ∩ 〈𝐵, 𝐴〉 |
10 | 2, 1 | op1stb 4394 | . 2 ⊢ ∩ ∩ 〈𝐵, 𝐴〉 = 𝐵 |
11 | 9, 10 | eqtri 2158 | 1 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 〈cop 3525 ∩ cint 3766 ◡ccnv 4533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-int 3767 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 |
This theorem is referenced by: 2ndval2 6047 |
Copyright terms: Public domain | W3C validator |