![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ssfidc | GIF version |
Description: A subset of a finite set is finite if membership in the subset is decidable. (Contributed by Jim Kingdon, 27-May-2022.) |
Ref | Expression |
---|---|
ssfidc | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss1 3354 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
2 | 1 | biimpi 120 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵) |
3 | 2 | 3ad2ant2 1021 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) = 𝐵) |
4 | dfin5 3151 | . . 3 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
5 | simp1 999 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐴 ∈ Fin) | |
6 | simp3 1001 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) | |
7 | 5, 6 | ssfirab 6961 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} ∈ Fin) |
8 | 4, 7 | eqeltrid 2276 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) ∈ Fin) |
9 | 3, 8 | eqeltrrd 2267 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 ∀wral 2468 {crab 2472 ∩ cin 3143 ⊆ wss 3144 Fincfn 6765 |
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-in1 615 ax-in2 616 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 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-1o 6440 df-er 6558 df-en 6766 df-fin 6768 |
This theorem is referenced by: infidc 6963 fisumss 11431 fprodssdc 11629 eulerthlemfi 12259 phisum 12271 sumhashdc 12378 1arith 12398 4sqlemafi 12426 |
Copyright terms: Public domain | W3C validator |