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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 5429 to extract the first member, op2nda 6181 for an alternate version, and op2nd 7931 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 6179 | . . . . . 6 ⊢ ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩} |
4 | 3 | inteqi 4912 | . . . . 5 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ {⟨𝐵, 𝐴⟩} |
5 | opex 5422 | . . . . . 6 ⊢ ⟨𝐵, 𝐴⟩ ∈ V | |
6 | 5 | intsn 4948 | . . . . 5 ⊢ ∩ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩ |
7 | 4, 6 | eqtri 2765 | . . . 4 ⊢ ∩ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩ |
8 | 7 | inteqi 4912 | . . 3 ⊢ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ⟨𝐵, 𝐴⟩ |
9 | 8 | inteqi 4912 | . 2 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = ∩ ∩ ⟨𝐵, 𝐴⟩ |
10 | 2, 1 | op1stb 5429 | . 2 ⊢ ∩ ∩ ⟨𝐵, 𝐴⟩ = 𝐵 |
11 | 9, 10 | eqtri 2765 | 1 ⊢ ∩ ∩ ∩ ◡{⟨𝐴, 𝐵⟩} = 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3446 {csn 4587 ⟨cop 4593 ∩ cint 4908 ◡ccnv 5633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-int 4909 df-br 5107 df-opab 5169 df-xp 5640 df-rel 5641 df-cnv 5642 |
This theorem is referenced by: 2ndval2 7940 |
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