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Mirrors > Home > ILE Home > Th. List > hashprg | Unicode 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 | ♯ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 525 | . . . . 5 | |
2 | snfig 6792 | . . . . . 6 | |
3 | 2 | ad2antrr 485 | . . . . 5 |
4 | elsni 3601 | . . . . . . . 8 | |
5 | 4 | eqcomd 2176 | . . . . . . 7 |
6 | 5 | necon3ai 2389 | . . . . . 6 |
7 | 6 | adantl 275 | . . . . 5 |
8 | hashunsng 10742 | . . . . . 6 ♯ ♯ | |
9 | 8 | imp 123 | . . . . 5 ♯ ♯ |
10 | 1, 3, 7, 9 | syl12anc 1231 | . . . 4 ♯ ♯ |
11 | hashsng 10733 | . . . . . . 7 ♯ | |
12 | 11 | adantr 274 | . . . . . 6 ♯ |
13 | 12 | adantr 274 | . . . . 5 ♯ |
14 | 13 | oveq1d 5868 | . . . 4 ♯ |
15 | 10, 14 | eqtrd 2203 | . . 3 ♯ |
16 | df-pr 3590 | . . . 4 | |
17 | 16 | fveq2i 5499 | . . 3 ♯ ♯ |
18 | df-2 8937 | . . 3 | |
19 | 15, 17, 18 | 3eqtr4g 2228 | . 2 ♯ |
20 | 1ne2 9084 | . . . . . . 7 | |
21 | 20 | a1i 9 | . . . . . 6 |
22 | 12, 21 | eqnetrd 2364 | . . . . 5 ♯ |
23 | dfsn2 3597 | . . . . . . . 8 | |
24 | preq2 3661 | . . . . . . . 8 | |
25 | 23, 24 | eqtr2id 2216 | . . . . . . 7 |
26 | 25 | fveq2d 5500 | . . . . . 6 ♯ ♯ |
27 | 26 | neeq1d 2358 | . . . . 5 ♯ ♯ |
28 | 22, 27 | syl5ibrcom 156 | . . . 4 ♯ |
29 | 28 | necon2d 2399 | . . 3 ♯ |
30 | 29 | imp 123 | . 2 ♯ |
31 | 19, 30 | impbida 591 | 1 ♯ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wne 2340 cun 3119 csn 3583 cpr 3584 cfv 5198 (class class class)co 5853 cfn 6718 c1 7775 caddc 7777 c2 8929 ♯chash 10709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-frec 6370 df-1o 6395 df-oadd 6399 df-er 6513 df-en 6719 df-dom 6720 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-ihash 10710 |
This theorem is referenced by: prhash2ex 10744 fiprsshashgt1 10752 |
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