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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 5200 | . . . . . 6 ⊢ ∅ ∈ V | |
2 | hashsng 13936 | . . . . . 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 488 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → (♯‘𝐴) = (♯‘{∅})) | |
8 | 1nn0 12106 | . . . . . . . . 9 ⊢ 1 ∈ ℕ0 | |
9 | 3, 8 | eqeltri 2834 | . . . . . . . 8 ⊢ (♯‘{∅}) ∈ ℕ0 |
10 | hashvnfin 13927 | . . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘{∅}) ∈ ℕ0) → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) | |
11 | 9, 10 | mpan2 691 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ∈ Fin)) |
12 | 11 | imp 410 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ∈ Fin) |
13 | snfi 8721 | . . . . . 6 ⊢ {∅} ∈ Fin | |
14 | hashen 13913 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ {∅} ∈ Fin) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) | |
15 | 12, 13, 14 | sylancl 589 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
16 | 7, 15 | mpbid 235 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) = (♯‘{∅})) → 𝐴 ≈ {∅}) |
17 | 16 | ex 416 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) → 𝐴 ≈ {∅})) |
18 | hasheni 13914 | . . 3 ⊢ (𝐴 ≈ {∅} → (♯‘𝐴) = (♯‘{∅})) | |
19 | 17, 18 | impbid1 228 | . 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = (♯‘{∅}) ↔ 𝐴 ≈ {∅})) |
20 | df1o2 8214 | . . . . 5 ⊢ 1o = {∅} | |
21 | 20 | eqcomi 2746 | . . . 4 ⊢ {∅} = 1o |
22 | 21 | breq2i 5061 | . . 3 ⊢ (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o) |
23 | 22 | a1i 11 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ≈ {∅} ↔ 𝐴 ≈ 1o)) |
24 | 6, 19, 23 | 3bitrd 308 | 1 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ 𝐴 ≈ 1o)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∅c0 4237 {csn 4541 class class class wbr 5053 ‘cfv 6380 1oc1o 8195 ≈ cen 8623 Fincfn 8626 1c1 10730 ℕ0cn0 12090 ♯chash 13896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-hash 13897 |
This theorem is referenced by: hash1elsn 13938 euhash1 13987 0ring 20308 0ring01eqbi 20311 lfuhgr3 32794 spthcycl 32804 |
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