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Mirrors > Home > MPE Home > Th. List > hash2prde | Structured version Visualization version GIF version |
Description: A set of size two is an unordered pair of two different elements. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
Ref | Expression |
---|---|
hash2prde | ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hash2pr 13637 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏}) | |
2 | equid 1970 | . . . . . . 7 ⊢ 𝑏 = 𝑏 | |
3 | vex 3413 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
4 | vex 3413 | . . . . . . . . 9 ⊢ 𝑏 ∈ V | |
5 | 3, 4 | preqsn 4663 | . . . . . . . 8 ⊢ ({𝑎, 𝑏} = {𝑏} ↔ (𝑎 = 𝑏 ∧ 𝑏 = 𝑏)) |
6 | eqeq2 2784 | . . . . . . . . . 10 ⊢ ({𝑎, 𝑏} = {𝑏} → (𝑉 = {𝑎, 𝑏} ↔ 𝑉 = {𝑏})) | |
7 | fveq2 6497 | . . . . . . . . . . . 12 ⊢ (𝑉 = {𝑏} → (♯‘𝑉) = (♯‘{𝑏})) | |
8 | hashsng 13543 | . . . . . . . . . . . . 13 ⊢ (𝑏 ∈ V → (♯‘{𝑏}) = 1) | |
9 | 8 | elv 3415 | . . . . . . . . . . . 12 ⊢ (♯‘{𝑏}) = 1 |
10 | 7, 9 | syl6eq 2825 | . . . . . . . . . . 11 ⊢ (𝑉 = {𝑏} → (♯‘𝑉) = 1) |
11 | eqeq1 2777 | . . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) = 2 → ((♯‘𝑉) = 1 ↔ 2 = 1)) | |
12 | 1ne2 11654 | . . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2 | |
13 | df-ne 2963 | . . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 2 ↔ ¬ 1 = 2) | |
14 | pm2.21 121 | . . . . . . . . . . . . . . . 16 ⊢ (¬ 1 = 2 → (1 = 2 → 𝑎 ≠ 𝑏)) | |
15 | 13, 14 | sylbi 209 | . . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → (1 = 2 → 𝑎 ≠ 𝑏)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . . . . . . 14 ⊢ (1 = 2 → 𝑎 ≠ 𝑏) |
17 | 16 | eqcoms 2781 | . . . . . . . . . . . . 13 ⊢ (2 = 1 → 𝑎 ≠ 𝑏) |
18 | 11, 17 | syl6bi 245 | . . . . . . . . . . . 12 ⊢ ((♯‘𝑉) = 2 → ((♯‘𝑉) = 1 → 𝑎 ≠ 𝑏)) |
19 | 18 | adantl 474 | . . . . . . . . . . 11 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ((♯‘𝑉) = 1 → 𝑎 ≠ 𝑏)) |
20 | 10, 19 | syl5com 31 | . . . . . . . . . 10 ⊢ (𝑉 = {𝑏} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑎 ≠ 𝑏)) |
21 | 6, 20 | syl6bi 245 | . . . . . . . . 9 ⊢ ({𝑎, 𝑏} = {𝑏} → (𝑉 = {𝑎, 𝑏} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑎 ≠ 𝑏))) |
22 | 21 | impcomd 403 | . . . . . . . 8 ⊢ ({𝑎, 𝑏} = {𝑏} → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)) |
23 | 5, 22 | sylbir 227 | . . . . . . 7 ⊢ ((𝑎 = 𝑏 ∧ 𝑏 = 𝑏) → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)) |
24 | 2, 23 | mpan2 679 | . . . . . 6 ⊢ (𝑎 = 𝑏 → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)) |
25 | ax-1 6 | . . . . . 6 ⊢ (𝑎 ≠ 𝑏 → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)) | |
26 | 24, 25 | pm2.61ine 3046 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏) |
27 | simpr 477 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑉 = {𝑎, 𝑏}) | |
28 | 26, 27 | jca 504 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) |
29 | 28 | ex 405 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → (𝑉 = {𝑎, 𝑏} → (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))) |
30 | 29 | 2eximdv 1879 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → (∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏} → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))) |
31 | 1, 30 | mpd 15 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏(𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∃wex 1743 ∈ wcel 2051 ≠ wne 2962 Vcvv 3410 {csn 4436 {cpr 4438 ‘cfv 6186 1c1 10335 2c2 11494 ♯chash 13504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-int 4747 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-1o 7904 df-2o 7905 df-oadd 7908 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-fin 8309 df-dju 9123 df-card 9161 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-n0 11707 df-z 11793 df-uz 12058 df-fz 12708 df-hash 13505 |
This theorem is referenced by: hash2exprb 13639 umgredg 26642 frgrregord013 27968 |
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