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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 3219 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∩ 𝐵) = 𝐵) | |
2 | 1 | biimpi 119 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵) |
3 | 2 | 3ad2ant2 968 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) = 𝐵) |
4 | dfin5 3020 | . . 3 ⊢ (𝐴 ∩ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} | |
5 | simp1 946 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐴 ∈ Fin) | |
6 | simp3 948 | . . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) | |
7 | 5, 6 | ssfirab 6723 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → {𝑥 ∈ 𝐴 ∣ 𝑥 ∈ 𝐵} ∈ Fin) |
8 | 4, 7 | syl5eqel 2181 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → (𝐴 ∩ 𝐵) ∈ Fin) |
9 | 3, 8 | eqeltrrd 2172 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 DECID 𝑥 ∈ 𝐵) → 𝐵 ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 783 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ∀wral 2370 {crab 2374 ∩ cin 3012 ⊆ wss 3013 Fincfn 6537 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-1o 6219 df-er 6332 df-en 6538 df-fin 6540 |
This theorem is referenced by: fisumss 10951 |
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