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Mirrors > Home > MPE Home > Th. List > hashen1 | Structured version Visualization version GIF version |
Description: A set has size 1 if and only if it is equinumerous to the ordinal 1. (Contributed by AV, 14-Apr-2019.) |
Ref | Expression |
---|---|
hashen1 | ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5235 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | hashsng 14082 | . . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (♯‘{∅}) = 1 |
4 | 3 | eqcomi 2749 | . . . 4 ⊢ 1 = (♯‘{∅}) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 1 = (♯‘{∅})) |
6 | 5 | eqeq2d 2751 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = (♯‘{∅}))) |
7 | simpr 485 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → (♯‘𝐴) = (♯‘{∅})) | |
8 | 1nn0 12249 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
9 | 3, 8 | eqeltri 2837 | . . . . . . . 8 ⊢ (♯‘{∅}) ∈ ℕ0 |
10 | hashvnfin 14073 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘{∅}) ∈ ℕ0) → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) | |
11 | 9, 10 | mpan2 688 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) |
12 | 11 | imp 407 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ∈ Fin) |
13 | snfi 8817 | . . . . . 6 ⊢ {∅} ∈ Fin | |
14 | hashen 14059 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {∅} ∈ Fin) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) | |
15 | 12, 13, 14 | sylancl 586 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
16 | 7, 15 | mpbid 231 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ≈ {∅}) |
17 | 16 | ex 413 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ≈ {∅})) |
18 | hasheni 14060 | . . 3 ⊢ (𝐴 ≈ {∅} → (♯‘𝐴) = (♯‘{∅})) | |
19 | 17, 18 | impbid1 224 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
20 | df1o2 8295 | . . . . 5 ⊢ 1o = {∅} | |
21 | 20 | eqcomi 2749 | . . . 4 ⊢ {∅} = 1o |
22 | 21 | breq2i 5087 | . . 3 ⊢ (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o) |
23 | 22 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o)) |
24 | 6, 19, 23 | 3bitrd 305 | 1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∅c0 4262 {csn 4567 class class class wbr 5079 ‘cfv 6432 1oc1o 8281 ≈ cen 8713 Fincfn 8716 1c1 10873 ℕ0cn0 12233 ♯chash 14042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-hash 14043 |
This theorem is referenced by: hash1elsn 14084 euhash1 14133 0ring 20539 0ring01eqbi 20542 lfuhgr3 33077 spthcycl 33087 |
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