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Mirrors > Home > MPE Home > Th. List > hash3tr | Structured version Visualization version GIF version |
Description: A set of size three is an unordered triple. (Contributed by Alexander van der Vekens, 13-Sep-2018.) |
Ref | Expression |
---|---|
hash3tr | ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 11768 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
2 | hashvnfin 13576 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 3 ∈ ℕ0) → ((♯‘𝑉) = 3 → 𝑉 ∈ Fin)) | |
3 | 1, 2 | mpan2 687 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 3 → 𝑉 ∈ Fin)) |
4 | 3 | imp 407 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → 𝑉 ∈ Fin) |
5 | hash3 13620 | . . . . . . . 8 ⊢ (♯‘3o) = 3 | |
6 | 5 | eqcomi 2804 | . . . . . . 7 ⊢ 3 = (♯‘3o) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑉 ∈ Fin → 3 = (♯‘3o)) |
8 | 7 | eqeq2d 2805 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 3 ↔ (♯‘𝑉) = (♯‘3o))) |
9 | 3onn 8122 | . . . . . . . 8 ⊢ 3o ∈ ω | |
10 | nnfi 8562 | . . . . . . . 8 ⊢ (3o ∈ ω → 3o ∈ Fin) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 3o ∈ Fin |
12 | hashen 13562 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 3o ∈ Fin) → ((♯‘𝑉) = (♯‘3o) ↔ 𝑉 ≈ 3o)) | |
13 | 11, 12 | mpan2 687 | . . . . . 6 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = (♯‘3o) ↔ 𝑉 ≈ 3o)) |
14 | 13 | biimpd 230 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = (♯‘3o) → 𝑉 ≈ 3o)) |
15 | 8, 14 | sylbid 241 | . . . 4 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 3 → 𝑉 ≈ 3o)) |
16 | 15 | adantld 491 | . . 3 ⊢ (𝑉 ∈ Fin → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → 𝑉 ≈ 3o)) |
17 | 4, 16 | mpcom 38 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → 𝑉 ≈ 3o) |
18 | en3 8606 | . 2 ⊢ (𝑉 ≈ 3o → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐}) | |
19 | 17, 18 | syl 17 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 3) → ∃𝑎∃𝑏∃𝑐 𝑉 = {𝑎, 𝑏, 𝑐}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 {ctp 4480 class class class wbr 4966 ‘cfv 6230 ωcom 7441 3oc3o 7953 ≈ cen 8359 Fincfn 8362 3c3 11546 ℕ0cn0 11750 ♯chash 13545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-1st 7550 df-2nd 7551 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-2o 7959 df-3o 7960 df-oadd 7962 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-dju 9181 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-hash 13546 |
This theorem is referenced by: hash1to3 13700 |
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