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Mirrors > Home > ILE Home > Th. List > fihashen1 | GIF version |
Description: A finite set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) (Intuitionized by Jim Kingdon, 23-Feb-2022.) |
Ref | Expression |
---|---|
fihashen1 | ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4025 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | hashsng 10512 | . . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (♯‘{∅}) = 1 |
4 | 3 | eqcomi 2121 | . . . 4 ⊢ 1 = (♯‘{∅}) |
5 | 4 | a1i 9 | . . 3 ⊢ (𝐴 ∈ Fin → 1 = (♯‘{∅})) |
6 | 5 | eqeq2d 2129 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = (♯‘{∅}))) |
7 | snfig 6676 | . . . 4 ⊢ (∅ ∈ V → {∅} ∈ Fin) | |
8 | 1, 7 | ax-mp 5 | . . 3 ⊢ {∅} ∈ Fin |
9 | hashen 10498 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ {∅} ∈ Fin) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) | |
10 | 8, 9 | mpan2 421 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
11 | df1o2 6294 | . . . . 5 ⊢ 1o = {∅} | |
12 | 11 | eqcomi 2121 | . . . 4 ⊢ {∅} = 1o |
13 | 12 | breq2i 3907 | . . 3 ⊢ (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o) |
14 | 13 | a1i 9 | . 2 ⊢ (𝐴 ∈ Fin → (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o)) |
15 | 6, 10, 14 | 3bitrd 213 | 1 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∅c0 3333 {csn 3497 class class class wbr 3899 ‘cfv 5093 1oc1o 6274 ≈ cen 6600 Fincfn 6602 1c1 7589 ♯chash 10489 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-recs 6170 df-frec 6256 df-1o 6281 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-inn 8689 df-n0 8946 df-z 9023 df-uz 9295 df-fz 9759 df-ihash 10490 |
This theorem is referenced by: (None) |
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