![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > snfig | GIF version |
Description: A singleton is finite. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Ref | Expression |
---|---|
snfig | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6209 | . . 3 ⊢ 1𝑜 ∈ ω | |
2 | ensn1g 6444 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) | |
3 | breq2 3815 | . . . 4 ⊢ (𝑥 = 1𝑜 → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1𝑜)) | |
4 | 3 | rspcev 2712 | . . 3 ⊢ ((1𝑜 ∈ ω ∧ {𝐴} ≈ 1𝑜) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
5 | 1, 2, 4 | sylancr 405 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
6 | isfi 6408 | . 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) | |
7 | 5, 6 | sylibr 132 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 ∃wrex 2354 {csn 3422 class class class wbr 3811 ωcom 4368 1𝑜c1o 6106 ≈ cen 6385 Fincfn 6387 |
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 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-1o 6113 df-en 6388 df-fin 6390 |
This theorem is referenced by: fiprc 6462 ssfiexmid 6522 domfiexmid 6524 diffitest 6533 unfiexmid 6555 prfidisj 6564 infpwfidom 6727 hashsng 10041 fihashen1 10042 hashunsng 10050 hashprg 10051 hashdifsn 10062 hashdifpr 10063 hashxp 10069 phicl2 10970 |
Copyright terms: Public domain | W3C validator |