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Mirrors > Home > MPE Home > Th. List > ex-hash | Structured version Visualization version GIF version |
Description: Example for df-hash 13694. (Contributed by AV, 4-Sep-2021.) |
Ref | Expression |
---|---|
ex-hash | ⊢ (♯‘{0, 1, 2}) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 4574 | . . . 4 ⊢ {0, 1, 2} = ({0, 1} ∪ {2}) | |
2 | 1 | fveq2i 6675 | . . 3 ⊢ (♯‘{0, 1, 2}) = (♯‘({0, 1} ∪ {2})) |
3 | prfi 8795 | . . . 4 ⊢ {0, 1} ∈ Fin | |
4 | snfi 8596 | . . . 4 ⊢ {2} ∈ Fin | |
5 | 2ne0 11744 | . . . . . 6 ⊢ 2 ≠ 0 | |
6 | 1ne2 11848 | . . . . . . 7 ⊢ 1 ≠ 2 | |
7 | 6 | necomi 3072 | . . . . . 6 ⊢ 2 ≠ 1 |
8 | 5, 7 | nelpri 4596 | . . . . 5 ⊢ ¬ 2 ∈ {0, 1} |
9 | disjsn 4649 | . . . . 5 ⊢ (({0, 1} ∩ {2}) = ∅ ↔ ¬ 2 ∈ {0, 1}) | |
10 | 8, 9 | mpbir 233 | . . . 4 ⊢ ({0, 1} ∩ {2}) = ∅ |
11 | hashun 13746 | . . . 4 ⊢ (({0, 1} ∈ Fin ∧ {2} ∈ Fin ∧ ({0, 1} ∩ {2}) = ∅) → (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2}))) | |
12 | 3, 4, 10, 11 | mp3an 1457 | . . 3 ⊢ (♯‘({0, 1} ∪ {2})) = ((♯‘{0, 1}) + (♯‘{2})) |
13 | 2, 12 | eqtri 2846 | . 2 ⊢ (♯‘{0, 1, 2}) = ((♯‘{0, 1}) + (♯‘{2})) |
14 | prhash2ex 13763 | . . . 4 ⊢ (♯‘{0, 1}) = 2 | |
15 | 2z 12017 | . . . . 5 ⊢ 2 ∈ ℤ | |
16 | hashsng 13733 | . . . . 5 ⊢ (2 ∈ ℤ → (♯‘{2}) = 1) | |
17 | 15, 16 | ax-mp 5 | . . . 4 ⊢ (♯‘{2}) = 1 |
18 | 14, 17 | oveq12i 7170 | . . 3 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = (2 + 1) |
19 | 2p1e3 11782 | . . 3 ⊢ (2 + 1) = 3 | |
20 | 18, 19 | eqtri 2846 | . 2 ⊢ ((♯‘{0, 1}) + (♯‘{2})) = 3 |
21 | 13, 20 | eqtri 2846 | 1 ⊢ (♯‘{0, 1, 2}) = 3 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 {csn 4569 {cpr 4571 {ctp 4573 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 0cc0 10539 1c1 10540 + caddc 10542 2c2 11695 3c3 11696 ℤcz 11984 ♯chash 13693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-hash 13694 |
This theorem is referenced by: (None) |
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