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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 5365 to extract the first member, op2nda 6087 for an alternate version, and op2nd 7700 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 6085 | . . . . . 6 ⊢ ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉} |
4 | 3 | inteqi 4882 | . . . . 5 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = ∩ {〈𝐵, 𝐴〉} |
5 | opex 5358 | . . . . . 6 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
6 | 5 | intsn 4914 | . . . . 5 ⊢ ∩ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
7 | 4, 6 | eqtri 2846 | . . . 4 ⊢ ∩ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
8 | 7 | inteqi 4882 | . . 3 ⊢ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ 〈𝐵, 𝐴〉 |
9 | 8 | inteqi 4882 | . 2 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = ∩ ∩ 〈𝐵, 𝐴〉 |
10 | 2, 1 | op1stb 5365 | . 2 ⊢ ∩ ∩ 〈𝐵, 𝐴〉 = 𝐵 |
11 | 9, 10 | eqtri 2846 | 1 ⊢ ∩ ∩ ∩ ◡{〈𝐴, 𝐵〉} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 〈cop 4575 ∩ cint 4878 ◡ccnv 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-int 4879 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: 2ndval2 7709 |
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