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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 14488 | . 2 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → ∃𝑎∃𝑏 𝑉 = {𝑎, 𝑏}) | |
2 | equid 2008 | . . . . . . 7 ⊢ 𝑏 = 𝑏 | |
3 | vex 3466 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
4 | vex 3466 | . . . . . . . . 9 ⊢ 𝑏 ∈ V | |
5 | 3, 4 | preqsn 4868 | . . . . . . . 8 ⊢ ({𝑎, 𝑏} = {𝑏} ↔ (𝑎 = 𝑏 ∧ 𝑏 = 𝑏)) |
6 | eqeq2 2738 | . . . . . . . . . 10 ⊢ ({𝑎, 𝑏} = {𝑏} → (𝑉 = {𝑎, 𝑏} ↔ 𝑉 = {𝑏})) | |
7 | fveq2 6901 | . . . . . . . . . . . 12 ⊢ (𝑉 = {𝑏} → (♯‘𝑉) = (♯‘{𝑏})) | |
8 | hashsng 14386 | . . . . . . . . . . . . 13 ⊢ (𝑏 ∈ V → (♯‘{𝑏}) = 1) | |
9 | 8 | elv 3468 | . . . . . . . . . . . 12 ⊢ (♯‘{𝑏}) = 1 |
10 | 7, 9 | eqtrdi 2782 | . . . . . . . . . . 11 ⊢ (𝑉 = {𝑏} → (♯‘𝑉) = 1) |
11 | eqeq1 2730 | . . . . . . . . . . . . 13 ⊢ ((♯‘𝑉) = 2 → ((♯‘𝑉) = 1 ↔ 2 = 1)) | |
12 | 1ne2 12472 | . . . . . . . . . . . . . . 15 ⊢ 1 ≠ 2 | |
13 | df-ne 2931 | . . . . . . . . . . . . . . . 16 ⊢ (1 ≠ 2 ↔ ¬ 1 = 2) | |
14 | pm2.21 123 | . . . . . . . . . . . . . . . 16 ⊢ (¬ 1 = 2 → (1 = 2 → 𝑎 ≠ 𝑏)) | |
15 | 13, 14 | sylbi 216 | . . . . . . . . . . . . . . 15 ⊢ (1 ≠ 2 → (1 = 2 → 𝑎 ≠ 𝑏)) |
16 | 12, 15 | ax-mp 5 | . . . . . . . . . . . . . 14 ⊢ (1 = 2 → 𝑎 ≠ 𝑏) |
17 | 16 | eqcoms 2734 | . . . . . . . . . . . . 13 ⊢ (2 = 1 → 𝑎 ≠ 𝑏) |
18 | 11, 17 | biimtrdi 252 | . . . . . . . . . . . 12 ⊢ ((♯‘𝑉) = 2 → ((♯‘𝑉) = 1 → 𝑎 ≠ 𝑏)) |
19 | 18 | adantl 480 | . . . . . . . . . . 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 | biimtrdi 252 | . . . . . . . . 9 ⊢ ({𝑎, 𝑏} = {𝑏} → (𝑉 = {𝑎, 𝑏} → ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → 𝑎 ≠ 𝑏))) |
22 | 21 | impcomd 410 | . . . . . . . 8 ⊢ ({𝑎, 𝑏} = {𝑏} → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)) |
23 | 5, 22 | sylbir 234 | . . . . . . 7 ⊢ ((𝑎 = 𝑏 ∧ 𝑏 = 𝑏) → (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏)) |
24 | 2, 23 | mpan2 689 | . . . . . 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 3015 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑎 ≠ 𝑏) |
27 | simpr 483 | . . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → 𝑉 = {𝑎, 𝑏}) | |
28 | 26, 27 | jca 510 | . . . 4 ⊢ (((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) ∧ 𝑉 = {𝑎, 𝑏}) → (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏})) |
29 | 28 | ex 411 | . . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ (♯‘𝑉) = 2) → (𝑉 = {𝑎, 𝑏} → (𝑎 ≠ 𝑏 ∧ 𝑉 = {𝑎, 𝑏}))) |
30 | 29 | 2eximdv 1915 | . 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 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 Vcvv 3462 {csn 4633 {cpr 4635 ‘cfv 6554 1c1 11159 2c2 12319 ♯chash 14347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-oadd 8500 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-dju 9944 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-hash 14348 |
This theorem is referenced by: hash2exprb 14490 umgredg 29074 frgrregord013 30328 |
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