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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 3331 | . . . . . 6 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦) | |
2 | 1 | biimpi 119 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦) |
3 | 2 | adantl 275 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦) |
4 | infiexmid.1 | . . . . 5 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin) | |
5 | 4 | adantr 274 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ Fin) |
6 | 3, 5 | eqeltrrd 2248 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
7 | 6 | gen2 1443 | . 2 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
8 | 7 | ssfiexmid 6852 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ∩ cin 3120 ⊆ wss 3121 Fincfn 6716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-br 3988 df-opab 4049 df-id 4276 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-1o 6393 df-er 6511 df-en 6717 df-fin 6719 |
This theorem is referenced by: (None) |
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