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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 3699 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}} |
4 | 3 | inteqi 3770 | . . . 4 ⊢ ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}} |
5 | 1 | snex 4104 | . . . . . 6 ⊢ {𝐴} ∈ V |
6 | prexg 4128 | . . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} ∈ V) | |
7 | 1, 2, 6 | mp2an 422 | . . . . . 6 ⊢ {𝐴, 𝐵} ∈ V |
8 | 5, 7 | intpr 3798 | . . . . 5 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵}) |
9 | snsspr1 3663 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
10 | df-ss 3079 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
11 | 9, 10 | mpbi 144 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
12 | 8, 11 | eqtri 2158 | . . . 4 ⊢ ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴} |
13 | 4, 12 | eqtri 2158 | . . 3 ⊢ ∩ 〈𝐴, 𝐵〉 = {𝐴} |
14 | 13 | inteqi 3770 | . 2 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴} |
15 | 1 | intsn 3801 | . 2 ⊢ ∩ {𝐴} = 𝐴 |
16 | 14, 15 | eqtri 2158 | 1 ⊢ ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2681 ∩ cin 3065 ⊆ wss 3066 {csn 3522 {cpr 3523 〈cop 3525 ∩ cint 3766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-int 3767 |
This theorem is referenced by: elreldm 4760 op2ndb 5017 1stval2 6046 fundmen 6693 xpsnen 6708 |
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