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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 5269 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | hashsng 14276 | . . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ (♯‘{∅}) = 1 |
4 | 3 | eqcomi 2746 | . . . 4 ⊢ 1 = (♯‘{∅}) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 1 = (♯‘{∅})) |
6 | 5 | eqeq2d 2748 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ (♯‘𝐴) = (♯‘{∅}))) |
7 | simpr 486 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → (♯‘𝐴) = (♯‘{∅})) | |
8 | 1nn0 12436 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
9 | 3, 8 | eqeltri 2834 | . . . . . . . 8 ⊢ (♯‘{∅}) ∈ ℕ0 |
10 | hashvnfin 14267 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘{∅}) ∈ ℕ0) → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) | |
11 | 9, 10 | mpan2 690 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) |
12 | 11 | imp 408 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ∈ Fin) |
13 | snfi 8995 | . . . . . 6 ⊢ {∅} ∈ Fin | |
14 | hashen 14254 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {∅} ∈ Fin) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) | |
15 | 12, 13, 14 | sylancl 587 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
16 | 7, 15 | mpbid 231 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ≈ {∅}) |
17 | 16 | ex 414 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ≈ {∅})) |
18 | hasheni 14255 | . . 3 ⊢ (𝐴 ≈ {∅} → (♯‘𝐴) = (♯‘{∅})) | |
19 | 17, 18 | impbid1 224 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
20 | df1o2 8424 | . . . . 5 ⊢ 1o = {∅} | |
21 | 20 | eqcomi 2746 | . . . 4 ⊢ {∅} = 1o |
22 | 21 | breq2i 5118 | . . 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 397 = wceq 1542 ∈ wcel 2107 Vcvv 3448 ∅c0 4287 {csn 4591 class class class wbr 5110 ‘cfv 6501 1oc1o 8410 ≈ cen 8887 Fincfn 8890 1c1 11059 ℕ0cn0 12420 ♯chash 14237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-fz 13432 df-hash 14238 |
This theorem is referenced by: hash1elsn 14278 euhash1 14327 0ring 20756 0ring01eqbi 20759 lfuhgr3 33753 spthcycl 33763 |
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