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Mirrors > Home > MPE Home > Th. List > hash2pr | Structured version Visualization version GIF version |
Description: A set of size two is an unordered pair. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
Ref | Expression |
---|---|
hash2pr | ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn0 11521 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | hashvnfin 13363 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 ∈ ℕ0) → ((♯‘𝑉) = 2 → 𝑉 ∈ Fin)) | |
3 | 1, 2 | mpan2 709 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 2 → 𝑉 ∈ Fin)) |
4 | 3 | imp 444 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑉 ∈ Fin) |
5 | hash2 13405 | . . . . . . . 8 ⊢ (♯‘2𝑜) = 2 | |
6 | 5 | eqcomi 2769 | . . . . . . 7 ⊢ 2 = (♯‘2𝑜) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑉 ∈ Fin → 2 = (♯‘2𝑜)) |
8 | 7 | eqeq2d 2770 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 2 ↔ (♯‘𝑉) = (♯‘2𝑜))) |
9 | 2onn 7891 | . . . . . . . 8 ⊢ 2𝑜 ∈ ω | |
10 | nnfi 8320 | . . . . . . . 8 ⊢ (2𝑜 ∈ ω → 2𝑜 ∈ Fin) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 2𝑜 ∈ Fin |
12 | hashen 13349 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 2𝑜 ∈ Fin) → ((♯‘𝑉) = (♯‘2𝑜) ↔ 𝑉 ≈ 2𝑜)) | |
13 | 11, 12 | mpan2 709 | . . . . . 6 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = (♯‘2𝑜) ↔ 𝑉 ≈ 2𝑜)) |
14 | 13 | biimpd 219 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = (♯‘2𝑜) → 𝑉 ≈ 2𝑜)) |
15 | 8, 14 | sylbid 230 | . . . 4 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 2 → 𝑉 ≈ 2𝑜)) |
16 | 15 | adantld 484 | . . 3 ⊢ (𝑉 ∈ Fin → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑉 ≈ 2𝑜)) |
17 | 4, 16 | mpcom 38 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑉 ≈ 2𝑜) |
18 | en2 8363 | . 2 ⊢ (𝑉 ≈ 2𝑜 → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏}) | |
19 | 17, 18 | syl 17 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∃wex 1853 ∈ wcel 2139 {cpr 4323 class class class wbr 4804 ‘cfv 6049 ωcom 7231 2𝑜c2o 7724 ≈ cen 8120 Fincfn 8123 2c2 11282 ℕ0cn0 11504 ♯chash 13331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-2o 7731 df-oadd 7734 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-card 8975 df-cda 9202 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-hash 13332 |
This theorem is referenced by: hash2prde 13464 hashle2pr 13471 hash1to3 13485 |
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