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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 6077 to extract the second member, op1sta 6075 for an alternate version, and op1st 7686 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 4794 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 4871 | . . . 4 ⊢ ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | snex 5322 | . . . . . 6 ⊢ {𝐴} ∈ V | |
6 | prex 5323 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V | |
7 | 5, 6 | intpr 4900 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
8 | snsspr1 4739 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
9 | df-ss 3949 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
10 | 8, 9 | mpbi 231 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
11 | 7, 10 | eqtri 2841 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
12 | 4, 11 | eqtri 2841 | . . 3 ⊢ ∩ 〈𝐴, 𝐵〉 = {𝐴} |
13 | 12 | inteqi 4871 | . 2 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴} |
14 | 1 | intsn 4903 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
15 | 13, 14 | eqtri 2841 | 1 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∩ cin 3932 ⊆ wss 3933 {csn 4557 {cpr 4559 〈cop 4563 ∩ cint 4867 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-int 4868 |
This theorem is referenced by: elreldm 5798 op2ndb 6077 elxp5 7617 1stval2 7695 fundmen 8571 xpsnen 8589 |
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