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Mirrors > Home > ILE Home > Th. List > snfig | GIF version |
Description: A singleton is finite. For the proper class case, see snprc 3535. (Contributed by Jim Kingdon, 13-Apr-2020.) |
Ref | Expression |
---|---|
snfig | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 6346 | . . 3 ⊢ 1o ∈ ω | |
2 | ensn1g 6621 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
3 | breq2 3879 | . . . 4 ⊢ (𝑥 = 1o → ({𝐴} ≈ 𝑥 ↔ {𝐴} ≈ 1o)) | |
4 | 3 | rspcev 2744 | . . 3 ⊢ ((1o ∈ ω ∧ {𝐴} ≈ 1o) → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
5 | 1, 2, 4 | sylancr 408 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 ∈ ω {𝐴} ≈ 𝑥) |
6 | isfi 6585 | . 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 1448 ∃wrex 2376 {csn 3474 class class class wbr 3875 ωcom 4442 1oc1o 6236 ≈ cen 6562 Fincfn 6564 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-id 4153 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-1o 6243 df-en 6565 df-fin 6567 |
This theorem is referenced by: fiprc 6639 ssfiexmid 6699 domfiexmid 6701 diffitest 6710 unfiexmid 6735 prfidisj 6744 tpfidisj 6745 infpwfidom 6963 hashsng 10385 fihashen1 10386 hashunsng 10394 hashprg 10395 hashdifsn 10406 hashdifpr 10407 hashxp 10413 fsumsplitsnun 11027 fsum2dlemstep 11042 fisumcom2 11046 fsumconst 11062 fsumge1 11069 fsum00 11070 hash2iun1dif1 11088 phicl2 11682 |
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