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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 5328 to extract the first member, op2nda 6052 for an alternate version, and op2nd 7680 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 6050 | . . . . . 6 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
4 | 3 | inteqi 4842 | . . . . 5 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = ∩ {〈𝐵, 𝐴〉} |
5 | opex 5321 | . . . . . 6 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
6 | 5 | intsn 4874 | . . . . 5 ⊢ ∩ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
7 | 4, 6 | eqtri 2821 | . . . 4 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
8 | 7 | inteqi 4842 | . . 3 ⊢ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ 〈𝐵, 𝐴〉 |
9 | 8 | inteqi 4842 | . 2 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ ∩ 〈𝐵, 𝐴〉 |
10 | 2, 1 | op1stb 5328 | . 2 ⊢ ∩ ∩ 〈𝐵, 𝐴〉 = 𝐵 |
11 | 9, 10 | eqtri 2821 | 1 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 {csn 4525 〈cop 4531 ∩ cint 4838 ◡ccnv 5518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-int 4839 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 |
This theorem is referenced by: 2ndval2 7689 |
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