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Mirrors > Home > MPE Home > Th. List > op1stb | Structured version Visualization version GIF version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (See op2ndb 6227 to extract the second member, op1sta 6225 for an alternate version, and op1st 7983 for the preferred version.) (Contributed by NM, 25-Nov-2003.) |
Ref | Expression |
---|---|
op1stb.1 | ⊢ 𝐴 ∈ V |
op1stb.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
op1stb | ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1stb.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | op1stb.2 | . . . . . 6 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | dfop 4873 | . . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 4955 | . . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 5432 | . . . . . 6 ⊢ {𝐴} ∈ V | |
6 | prex 5433 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | intpr 4987 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
8 | snsspr1 4818 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | df-ss 3966 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
10 | 8, 9 | mpbi 229 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
11 | 7, 10 | eqtri 2761 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
12 | 4, 11 | eqtri 2761 | . . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴} |
13 | 12 | inteqi 4955 | . 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴} |
14 | 1 | intsn 4991 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
15 | 13, 14 | eqtri 2761 | 1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 {csn 4629 {cpr 4631 ⟨cop 4635 ∩ cint 4951 |
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 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-int 4952 |
This theorem is referenced by: elreldm 5935 op2ndb 6227 elxp5 7914 1stval2 7992 fundmen 9031 xpsnen 9055 |
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