Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > snfig | GIF version |
Description: A singleton is finite. For the proper class case, see snprc 3583. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Ref | Expression |
---|---|
snfig | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6409 | . . 3 ⊢ 1o ∈ ω | |
2 | ensn1g 6684 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
3 | breq2 3928 | . . . 4 ⊢ (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o)) | |
4 | 3 | rspcev 2784 | . . 3 ⊢ ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
5 | 1, 2, 4 | sylancr 410 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
6 | isfi 6648 | . 2 ⊢ ({𝐴} ∈ Fin ↔ ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) | |
7 | 5, 6 | sylibr 133 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∃wrex 2415 {csn 3522 class class class wbr 3924 ωcom 4499 1oc1o 6299 ≈ cen 6625 Fincfn 6627 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-1o 6306 df-en 6628 df-fin 6630 |
This theorem is referenced by: fiprc 6702 ssfiexmid 6763 domfiexmid 6765 diffitest 6774 unfiexmid 6799 prfidisj 6808 tpfidisj 6809 ssfii 6855 infpwfidom 7047 hashsng 10537 fihashen1 10538 hashunsng 10546 hashprg 10547 hashdifsn 10558 hashdifpr 10559 hashxp 10565 fsumsplitsnun 11181 fsum2dlemstep 11196 fisumcom2 11200 fsumconst 11216 fsumge1 11223 fsum00 11224 hash2iun1dif1 11242 phicl2 11879 |
Copyright terms: Public domain | W3C validator |