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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 6225 to extract the second member, op1sta 6223 for an alternate version, and op1st 7985 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 4871 | . . . . 5 ⊢ ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 4953 | . . . 4 ⊢ ∩ ⟨𝐴, 𝐵⟩ = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 5430 | . . . . . 6 ⊢ {𝐴} ∈ V | |
6 | prex 5431 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | intpr 4985 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
8 | snsspr1 4816 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | df-ss 3964 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
10 | 8, 9 | mpbi 229 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
11 | 7, 10 | eqtri 2758 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
12 | 4, 11 | eqtri 2758 | . . 3 ⊢ ∩ ⟨𝐴, 𝐵⟩ = {𝐴} |
13 | 12 | inteqi 4953 | . 2 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = ∩ {𝐴} |
14 | 1 | intsn 4989 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
15 | 13, 14 | eqtri 2758 | 1 ⊢ ∩ ∩ ⟨𝐴, 𝐵⟩ = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2104 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 {csn 4627 {cpr 4629 ⟨cop 4633 ∩ cint 4949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-int 4950 |
This theorem is referenced by: elreldm 5933 op2ndb 6225 elxp5 7916 1stval2 7994 fundmen 9033 xpsnen 9057 |
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