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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 3190 | . . . . . 6 ⊢ (𝑦 ⊆ 𝑥 ↔ (𝑥 ∩ 𝑦) = 𝑦) | |
2 | 1 | biimpi 118 | . . . . 5 ⊢ (𝑦 ⊆ 𝑥 → (𝑥 ∩ 𝑦) = 𝑦) |
3 | 2 | adantl 271 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) = 𝑦) |
4 | infiexmid.1 | . . . . 5 ⊢ (𝑥 ∈ Fin → (𝑥 ∩ 𝑦) ∈ Fin) | |
5 | 4 | adantr 270 | . . . 4 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → (𝑥 ∩ 𝑦) ∈ Fin) |
6 | 3, 5 | eqeltrrd 2162 | . . 3 ⊢ ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
7 | 6 | gen2 1382 | . 2 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
8 | 7 | ssfiexmid 6525 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 662 = wceq 1287 ∈ wcel 1436 ∩ cin 2985 ⊆ wss 2986 Fincfn 6390 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-iinf 4369 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-rab 2364 df-v 2616 df-sbc 2829 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-br 3815 df-opab 3869 df-id 4087 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-1o 6116 df-er 6225 df-en 6391 df-fin 6393 |
This theorem is referenced by: (None) |
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