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Mirrors > Home > ILE Home > Th. List > op1stb | GIF version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (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 3589 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 3660 | . . . 4 ⊢ ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | 1 | snex 3977 | . . . . . 6 ⊢ {𝐴} ∈ V |
6 | prexg 3994 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 2, 6 | mp2an 417 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V |
8 | 5, 7 | intpr 3688 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
9 | snsspr1 3553 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
10 | df-ss 2995 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
11 | 9, 10 | mpbi 143 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
12 | 8, 11 | eqtri 2103 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
13 | 4, 12 | eqtri 2103 | . . 3 ⊢ ∩ 〈𝐴, 𝐵〉 = {𝐴} |
14 | 13 | inteqi 3660 | . 2 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴} |
15 | 1 | intsn 3691 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
16 | 14, 15 | eqtri 2103 | 1 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1285 ∈ wcel 1434 Vcvv 2610 ∩ cin 2981 ⊆ wss 2982 {csn 3416 {cpr 3417 〈cop 3419 ∩ cint 3656 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-v 2612 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-int 3657 |
This theorem is referenced by: elreldm 4608 op2ndb 4854 1stval2 5834 fundmen 6375 xpsnen 6387 |
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