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Mirrors > Home > MPE Home > Th. List > hash1snb | Structured version Visualization version GIF version |
Description: The size of a set is 1 if and only if it is a singleton (containing a set). (Contributed by Alexander van der Vekens, 7-Dec-2017.) |
Ref | Expression |
---|---|
hash1snb | ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . . . . 9 ⊢ ((♯‘𝑉) = 1 → (♯‘𝑉) = 1) | |
2 | hash1 14311 | . . . . . . . . 9 ⊢ (♯‘1o) = 1 | |
3 | 1, 2 | eqtr4di 2795 | . . . . . . . 8 ⊢ ((♯‘𝑉) = 1 → (♯‘𝑉) = (♯‘1o)) |
4 | 3 | adantl 483 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → (♯‘𝑉) = (♯‘1o)) |
5 | 1onn 8591 | . . . . . . . . 9 ⊢ 1o ∈ ω | |
6 | nnfi 9118 | . . . . . . . . 9 ⊢ (1o ∈ ω → 1o ∈ Fin) | |
7 | 5, 6 | mp1i 13 | . . . . . . . 8 ⊢ ((♯‘𝑉) = 1 → 1o ∈ Fin) |
8 | hashen 14254 | . . . . . . . 8 ⊢ ((𝑉 ∈ Fin ∧ 1o ∈ Fin) → ((♯‘𝑉) = (♯‘1o) ↔ 𝑉 ≈ 1o)) | |
9 | 7, 8 | sylan2 594 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → ((♯‘𝑉) = (♯‘1o) ↔ 𝑉 ≈ 1o)) |
10 | 4, 9 | mpbid 231 | . . . . . 6 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → 𝑉 ≈ 1o) |
11 | en1 8972 | . . . . . 6 ⊢ (𝑉 ≈ 1o ↔ ∃𝑎 𝑉 = {𝑎}) | |
12 | 10, 11 | sylib 217 | . . . . 5 ⊢ ((𝑉 ∈ Fin ∧ (♯‘𝑉) = 1) → ∃𝑎 𝑉 = {𝑎}) |
13 | 12 | ex 414 | . . . 4 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎})) |
14 | 13 | a1d 25 | . . 3 ⊢ (𝑉 ∈ Fin → (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎}))) |
15 | hashinf 14242 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ 𝑉 ∈ Fin) → (♯‘𝑉) = +∞) | |
16 | eqeq1 2741 | . . . . . 6 ⊢ ((♯‘𝑉) = +∞ → ((♯‘𝑉) = 1 ↔ +∞ = 1)) | |
17 | 1re 11162 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
18 | renepnf 11210 | . . . . . . . 8 ⊢ (1 ∈ ℝ → 1 ≠ +∞) | |
19 | df-ne 2945 | . . . . . . . . 9 ⊢ (1 ≠ +∞ ↔ ¬ 1 = +∞) | |
20 | pm2.21 123 | . . . . . . . . 9 ⊢ (¬ 1 = +∞ → (1 = +∞ → ∃𝑎 𝑉 = {𝑎})) | |
21 | 19, 20 | sylbi 216 | . . . . . . . 8 ⊢ (1 ≠ +∞ → (1 = +∞ → ∃𝑎 𝑉 = {𝑎})) |
22 | 17, 18, 21 | mp2b 10 | . . . . . . 7 ⊢ (1 = +∞ → ∃𝑎 𝑉 = {𝑎}) |
23 | 22 | eqcoms 2745 | . . . . . 6 ⊢ (+∞ = 1 → ∃𝑎 𝑉 = {𝑎}) |
24 | 16, 23 | syl6bi 253 | . . . . 5 ⊢ ((♯‘𝑉) = +∞ → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎})) |
25 | 15, 24 | syl 17 | . . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ ¬ 𝑉 ∈ Fin) → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎})) |
26 | 25 | expcom 415 | . . 3 ⊢ (¬ 𝑉 ∈ Fin → (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎}))) |
27 | 14, 26 | pm2.61i 182 | . 2 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 → ∃𝑎 𝑉 = {𝑎})) |
28 | fveq2 6847 | . . . 4 ⊢ (𝑉 = {𝑎} → (♯‘𝑉) = (♯‘{𝑎})) | |
29 | hashsng 14276 | . . . . 5 ⊢ (𝑎 ∈ V → (♯‘{𝑎}) = 1) | |
30 | 29 | elv 3454 | . . . 4 ⊢ (♯‘{𝑎}) = 1 |
31 | 28, 30 | eqtrdi 2793 | . . 3 ⊢ (𝑉 = {𝑎} → (♯‘𝑉) = 1) |
32 | 31 | exlimiv 1934 | . 2 ⊢ (∃𝑎 𝑉 = {𝑎} → (♯‘𝑉) = 1) |
33 | 27, 32 | impbid1 224 | 1 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 1 ↔ ∃𝑎 𝑉 = {𝑎})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2944 Vcvv 3448 {csn 4591 class class class wbr 5110 ‘cfv 6501 ωcom 7807 1oc1o 8410 ≈ cen 8887 Fincfn 8890 ℝcr 11057 1c1 11059 +∞cpnf 11193 ♯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-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-dju 9844 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: hash1n0 14328 hashle2pr 14383 hashge2el2difr 14387 hash1to3 14397 cshwrepswhash1 16982 symgvalstruct 19185 symgvalstructOLD 19186 mat1scmat 21904 tgldim0eq 27487 lfuhgr1v0e 28244 usgr1v0e 28316 nbgr1vtx 28348 uvtx01vtx 28387 cplgr1vlem 28419 cplgr1v 28420 1loopgrvd2 28493 vdgn1frgrv2 29282 frgrwopreg1 29304 frgrwopreg2 29305 extdg1id 32392 c0snmgmhm 46286 |
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