![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > infiexmid | GIF version |
Description: If the intersection of any finite set and any other set is finite, excluded middle follows. (Contributed by Jim Kingdon, 5-Feb-2022.) |
Ref | Expression |
---|---|
infiexmid.1 | ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin) |
Ref | Expression |
---|---|
infiexmid | ⊢ (𝜑 ∨ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss1 3219 | . . . . . 6 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦) | |
2 | 1 | biimpi 119 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦) |
3 | 2 | adantl 272 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦) |
4 | infiexmid.1 | . . . . 5 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin) | |
5 | 4 | adantr 271 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ Fin) |
6 | 3, 5 | eqeltrrd 2172 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
7 | 6 | gen2 1391 | . 2 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
8 | 7 | ssfiexmid 6672 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 667 = wceq 1296 ∈ wcel 1445 ∩ 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-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 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-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-id 4144 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: (None) |
Copyright terms: Public domain | W3C validator |