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Mirrors > Home > ILE Home > Th. List > apsscn | GIF version |
Description: The points apart from a given point are complex numbers. (Contributed by Jim Kingdon, 19-Dec-2023.) |
Ref | Expression |
---|---|
apsscn | ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 4032 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 # 𝐵 ↔ 𝑦 # 𝐵)) | |
2 | 1 | elrab 2916 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 # 𝐵)) |
3 | aprcl 8665 | . . . 4 ⊢ (𝑦 # 𝐵 → (𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ)) | |
4 | 2, 3 | simplbiim 387 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} → (𝑦 ∈ ℂ ∧ 𝐵 ∈ ℂ)) |
5 | 4 | simpld 112 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} → 𝑦 ∈ ℂ) |
6 | 5 | ssriv 3183 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝑥 # 𝐵} ⊆ ℂ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ∈ wcel 2164 {crab 2476 ⊆ wss 3153 class class class wbr 4029 ℂcc 7870 # cap 8600 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-resscn 7964 ax-icn 7967 ax-addcl 7968 ax-mulcl 7970 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fo 5260 df-fv 5262 df-1st 6193 df-2nd 6194 df-ap 8601 |
This theorem is referenced by: expghmap 14095 maxcncf 14769 mincncf 14770 limccoap 14832 dveflem 14872 |
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