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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 12485 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
2 | hashvnfin 14316 | . . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 2 ∈ ℕ0) → ((♯‘𝑉) = 2 → 𝑉 ∈ Fin)) | |
3 | 1, 2 | mpan2 689 | . . . 4 ⊢ (𝑉 ∈ 𝑊 → ((♯‘𝑉) = 2 → 𝑉 ∈ Fin)) |
4 | 3 | imp 407 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑉 ∈ Fin) |
5 | hash2 14361 | . . . . . . . 8 ⊢ (♯‘2o) = 2 | |
6 | 5 | eqcomi 2741 | . . . . . . 7 ⊢ 2 = (♯‘2o) |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝑉 ∈ Fin → 2 = (♯‘2o)) |
8 | 7 | eqeq2d 2743 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 2 ↔ (♯‘𝑉) = (♯‘2o))) |
9 | 2onn 8637 | . . . . . . . 8 ⊢ 2o ∈ ω | |
10 | nnfi 9163 | . . . . . . . 8 ⊢ (2o ∈ ω → 2o ∈ Fin) | |
11 | 9, 10 | ax-mp 5 | . . . . . . 7 ⊢ 2o ∈ Fin |
12 | hashen 14303 | . . . . . . 7 ⊢ ((𝑉 ∈ Fin ∧ 2o ∈ Fin) → ((♯‘𝑉) = (♯‘2o) ↔ 𝑉 ≈ 2o)) | |
13 | 11, 12 | mpan2 689 | . . . . . 6 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = (♯‘2o) ↔ 𝑉 ≈ 2o)) |
14 | 13 | biimpd 228 | . . . . 5 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = (♯‘2o) → 𝑉 ≈ 2o)) |
15 | 8, 14 | sylbid 239 | . . . 4 ⊢ (𝑉 ∈ Fin → ((♯‘𝑉) = 2 → 𝑉 ≈ 2o)) |
16 | 15 | adantld 491 | . . 3 ⊢ (𝑉 ∈ Fin → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑉 ≈ 2o)) |
17 | 4, 16 | mpcom 38 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑉 ≈ 2o) |
18 | en2 9277 | . 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 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 {cpr 4629 class class class wbr 5147 ‘cfv 6540 ωcom 7851 2oc2o 8456 ≈ cen 8932 Fincfn 8935 2c2 12263 ℕ0cn0 12468 ♯chash 14286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-dju 9892 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-hash 14287 |
This theorem is referenced by: hash2prde 14427 hashle2pr 14434 hash1to3 14448 |
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