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Mirrors > Home > MPE Home > Th. List > hashprg | Structured version Visualization version GIF version |
Description: The size of an unordered pair. (Contributed by Mario Carneiro, 27-Sep-2013.) (Revised by Mario Carneiro, 5-May-2016.) (Revised by AV, 18-Sep-2021.) |
Ref | Expression |
---|---|
hashprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 484 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
2 | elsni 4647 | . . . . . . 7 ⊢ (𝐵 ∈ {𝐴} → 𝐵 = 𝐴) | |
3 | 2 | eqcomd 2740 | . . . . . 6 ⊢ (𝐵 ∈ {𝐴} → 𝐴 = 𝐵) |
4 | 3 | necon3ai 2962 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ¬ 𝐵 ∈ {𝐴}) |
5 | snfi 9081 | . . . . . 6 ⊢ {𝐴} ∈ Fin | |
6 | hashunsng 14427 | . . . . . . 7 ⊢ (𝐵 ∈ 𝑊 → (({𝐴} ∈ Fin ∧ ¬ 𝐵 ∈ {𝐴}) → (♯‘({𝐴} ∪ {𝐵})) = ((♯‘{𝐴}) + 1))) | |
7 | 6 | imp 406 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑊 ∧ ({𝐴} ∈ Fin ∧ ¬ 𝐵 ∈ {𝐴})) → (♯‘({𝐴} ∪ {𝐵})) = ((♯‘{𝐴}) + 1)) |
8 | 5, 7 | mpanr1 703 | . . . . 5 ⊢ ((𝐵 ∈ 𝑊 ∧ ¬ 𝐵 ∈ {𝐴}) → (♯‘({𝐴} ∪ {𝐵})) = ((♯‘{𝐴}) + 1)) |
9 | 1, 4, 8 | syl2an 596 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → (♯‘({𝐴} ∪ {𝐵})) = ((♯‘{𝐴}) + 1)) |
10 | hashsng 14404 | . . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (♯‘{𝐴}) = 1) | |
11 | 10 | adantr 480 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (♯‘{𝐴}) = 1) |
12 | 11 | adantr 480 | . . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → (♯‘{𝐴}) = 1) |
13 | 12 | oveq1d 7445 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → ((♯‘{𝐴}) + 1) = (1 + 1)) |
14 | 9, 13 | eqtrd 2774 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → (♯‘({𝐴} ∪ {𝐵})) = (1 + 1)) |
15 | df-pr 4633 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
16 | 15 | fveq2i 6909 | . . 3 ⊢ (♯‘{𝐴, 𝐵}) = (♯‘({𝐴} ∪ {𝐵})) |
17 | df-2 12326 | . . 3 ⊢ 2 = (1 + 1) | |
18 | 14, 16, 17 | 3eqtr4g 2799 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐴 ≠ 𝐵) → (♯‘{𝐴, 𝐵}) = 2) |
19 | 1ne2 12471 | . . . . . . 7 ⊢ 1 ≠ 2 | |
20 | 19 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 1 ≠ 2) |
21 | 11, 20 | eqnetrd 3005 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (♯‘{𝐴}) ≠ 2) |
22 | dfsn2 4643 | . . . . . . . 8 ⊢ {𝐴} = {𝐴, 𝐴} | |
23 | preq2 4738 | . . . . . . . 8 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) | |
24 | 22, 23 | eqtr2id 2787 | . . . . . . 7 ⊢ (𝐴 = 𝐵 → {𝐴, 𝐵} = {𝐴}) |
25 | 24 | fveq2d 6910 | . . . . . 6 ⊢ (𝐴 = 𝐵 → (♯‘{𝐴, 𝐵}) = (♯‘{𝐴})) |
26 | 25 | neeq1d 2997 | . . . . 5 ⊢ (𝐴 = 𝐵 → ((♯‘{𝐴, 𝐵}) ≠ 2 ↔ (♯‘{𝐴}) ≠ 2)) |
27 | 21, 26 | syl5ibrcom 247 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 = 𝐵 → (♯‘{𝐴, 𝐵}) ≠ 2)) |
28 | 27 | necon2d 2960 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((♯‘{𝐴, 𝐵}) = 2 → 𝐴 ≠ 𝐵)) |
29 | 28 | imp 406 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (♯‘{𝐴, 𝐵}) = 2) → 𝐴 ≠ 𝐵) |
30 | 18, 29 | impbida 801 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐵 ↔ (♯‘{𝐴, 𝐵}) = 2)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∪ cun 3960 {csn 4630 {cpr 4632 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 1c1 11153 + caddc 11155 2c2 12318 ♯chash 14365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-dju 9938 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-hash 14366 |
This theorem is referenced by: hashprb 14432 prhash2ex 14434 hashfun 14472 hash2exprb 14506 nehash2 14509 hashtpg 14520 elss2prb 14523 hash3tpexb 14529 wrdlen2i 14977 isnzr2hash 20535 upgrex 29123 umgrbi 29132 usgr1e 29276 usgrexmplef 29290 cusgrexilem2 29473 cusgrfilem1 29487 umgr2v2e 29557 vdegp1bi 29569 eulerpathpr 30268 drngidlhash 33441 ccfldextdgrr 33696 coinflipprob 34460 cusgredgex 35105 subfacp1lem1 35163 poimirlem9 37615 fourierdlem54 46115 fourierdlem102 46163 fourierdlem103 46164 fourierdlem104 46165 fourierdlem114 46175 prpair 47425 prproropf1olem1 47427 paireqne 47435 prprspr2 47442 reuprpr 47447 stgrusgra 47861 gpgusgralem 47945 |
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