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Mirrors > Home > ILE Home > Th. List > op1stbg | GIF version |
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.) |
Ref | Expression |
---|---|
op1stbg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopg 3776 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 = {{𝐴}, {𝐴, 𝐵}}) | |
2 | 1 | inteqd 3849 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 〈𝐴, 𝐵〉 = ∩ {{𝐴}, {𝐴, 𝐵}}) |
3 | snexg 4184 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ V) | |
4 | prexg 4211 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ∈ V) | |
5 | intprg 3877 | . . . . . 6 ⊢ (({𝐴} ∈ V ∧ {𝐴, 𝐵} ∈ V) → ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})) | |
6 | 3, 4, 5 | syl2an2r 595 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {{𝐴}, {𝐴, 𝐵}} = ({𝐴} ∩ {𝐴, 𝐵})) |
7 | snsspr1 3740 | . . . . . 6 ⊢ {𝐴} ⊆ {𝐴, 𝐵} | |
8 | df-ss 3142 | . . . . . 6 ⊢ ({𝐴} ⊆ {𝐴, 𝐵} ↔ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴}) | |
9 | 7, 8 | mpbi 145 | . . . . 5 ⊢ ({𝐴} ∩ {𝐴, 𝐵}) = {𝐴} |
10 | 6, 9 | eqtrdi 2226 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {{𝐴}, {𝐴, 𝐵}} = {𝐴}) |
11 | 2, 10 | eqtrd 2210 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ 〈𝐴, 𝐵〉 = {𝐴}) |
12 | 11 | inteqd 3849 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ 〈𝐴, 𝐵〉 = ∩ {𝐴}) |
13 | intsng 3878 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∩ {𝐴} = 𝐴) | |
14 | 13 | adantr 276 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ {𝐴} = 𝐴) |
15 | 12, 14 | eqtrd 2210 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∩ ∩ 〈𝐴, 𝐵〉 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ∩ cin 3128 ⊆ wss 3129 {csn 3592 {cpr 3593 〈cop 3595 ∩ cint 3844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-int 3845 |
This theorem is referenced by: elxp5 5117 fundmen 6805 |
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