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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 6093 | . . 3 ⊢ 1𝑜 ∈ ω | |
2 | ensn1g 6277 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) | |
3 | breq2 3768 | . . . 4 ⊢ (𝑥 = 1𝑜 → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1𝑜)) | |
4 | 3 | rspcev 2656 | . . 3 ⊢ ((1𝑜 ∈ ω ∧ {𝐴} ≈ 1𝑜) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
5 | 1, 2, 4 | sylancr 393 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
6 | isfi 6241 | . 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) | |
7 | 5, 6 | sylibr 137 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ∃wrex 2307 {csn 3375 class class class wbr 3764 ωcom 4313 1𝑜c1o 5994 ≈ cen 6219 Fincfn 6221 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-1o 6001 df-en 6222 df-fin 6224 |
This theorem is referenced by: fiprc 6292 ssfiexmid 6336 diffitest 6344 |
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