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Mirrors > Home > ILE Home > Th. List > hashsng | GIF version |
Description: The size of a singleton. (Contributed by Paul Chapman, 26-Oct-2012.) (Proof shortened by Mario Carneiro, 13-Feb-2013.) |
Ref | Expression |
---|---|
hashsng | ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1z 9080 | . . . 4 ⊢ 1 ∈ ℤ | |
2 | en2sn 6707 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 ∈ ℤ) → {𝐴} ≈ {1}) | |
3 | 1, 2 | mpan2 421 | . . 3 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ {1}) |
4 | snfig 6708 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ∈ Fin) | |
5 | snfig 6708 | . . . . 5 ⊢ (1 ∈ ℤ → {1} ∈ Fin) | |
6 | 1, 5 | ax-mp 5 | . . . 4 ⊢ {1} ∈ Fin |
7 | hashen 10530 | . . . 4 ⊢ (({𝐴} ∈ Fin ∧ {1} ∈ Fin) → ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1})) | |
8 | 4, 6, 7 | sylancl 409 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘{𝐴}) = (♯‘{1}) ↔ {𝐴} ≈ {1})) |
9 | 3, 8 | mpbird 166 | . 2 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = (♯‘{1})) |
10 | 1nn0 8993 | . . . . 5 ⊢ 1 ∈ ℕ0 | |
11 | hashfz1 10529 | . . . . 5 ⊢ (1 ∈ ℕ0 → (♯‘(1...1)) = 1) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ (♯‘(1...1)) = 1 |
13 | fzsn 9846 | . . . . 5 ⊢ (1 ∈ ℤ → (1...1) = {1}) | |
14 | 13 | fveq2d 5425 | . . . 4 ⊢ (1 ∈ ℤ → (♯‘(1...1)) = (♯‘{1})) |
15 | 12, 14 | syl5reqr 2187 | . . 3 ⊢ (1 ∈ ℤ → (♯‘{1}) = 1) |
16 | 1, 15 | ax-mp 5 | . 2 ⊢ (♯‘{1}) = 1 |
17 | 9, 16 | syl6eq 2188 | 1 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 {csn 3527 class class class wbr 3929 ‘cfv 5123 (class class class)co 5774 ≈ cen 6632 Fincfn 6634 1c1 7621 ℕ0cn0 8977 ℤcz 9054 ...cfz 9790 ♯chash 10521 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-1o 6313 df-er 6429 df-en 6635 df-dom 6636 df-fin 6637 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 df-ihash 10522 |
This theorem is referenced by: fihashen1 10545 hashunsng 10553 hashprg 10554 hashdifsn 10565 fsumconst 11223 phicl2 11890 dfphi2 11896 |
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