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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 3354 | . . . . . 6 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦) | |
2 | 1 | biimpi 120 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦) |
3 | 2 | adantl 277 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦) |
4 | infiexmid.1 | . . . . 5 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin) | |
5 | 4 | adantr 276 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ Fin) |
6 | 3, 5 | eqeltrrd 2267 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
7 | 6 | gen2 1461 | . 2 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
8 | 7 | ssfiexmid 6903 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 709 = wceq 1364 ∈ wcel 2160 ∩ 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-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 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-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 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: (None) |
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