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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 519 | . . . . 5 | |
2 | snfig 6701 | . . . . . 6 | |
3 | 2 | ad2antrr 479 | . . . . 5 |
4 | elsni 3540 | . . . . . . . 8 | |
5 | 4 | eqcomd 2143 | . . . . . . 7 |
6 | 5 | necon3ai 2355 | . . . . . 6 |
7 | 6 | adantl 275 | . . . . 5 |
8 | hashunsng 10546 | . . . . . 6 ♯ ♯ | |
9 | 8 | imp 123 | . . . . 5 ♯ ♯ |
10 | 1, 3, 7, 9 | syl12anc 1214 | . . . 4 ♯ ♯ |
11 | hashsng 10537 | . . . . . . 7 ♯ | |
12 | 11 | adantr 274 | . . . . . 6 ♯ |
13 | 12 | adantr 274 | . . . . 5 ♯ |
14 | 13 | oveq1d 5782 | . . . 4 ♯ |
15 | 10, 14 | eqtrd 2170 | . . 3 ♯ |
16 | df-pr 3529 | . . . 4 | |
17 | 16 | fveq2i 5417 | . . 3 ♯ ♯ |
18 | df-2 8772 | . . 3 | |
19 | 15, 17, 18 | 3eqtr4g 2195 | . 2 ♯ |
20 | 1ne2 8919 | . . . . . . 7 | |
21 | 20 | a1i 9 | . . . . . 6 |
22 | 12, 21 | eqnetrd 2330 | . . . . 5 ♯ |
23 | dfsn2 3536 | . . . . . . . 8 | |
24 | preq2 3596 | . . . . . . . 8 | |
25 | 23, 24 | syl5req 2183 | . . . . . . 7 |
26 | 25 | fveq2d 5418 | . . . . . 6 ♯ ♯ |
27 | 26 | neeq1d 2324 | . . . . 5 ♯ ♯ |
28 | 22, 27 | syl5ibrcom 156 | . . . 4 ♯ |
29 | 28 | necon2d 2365 | . . 3 ♯ |
30 | 29 | imp 123 | . 2 ♯ |
31 | 19, 30 | impbida 585 | 1 ♯ |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wne 2306 cun 3064 csn 3522 cpr 3523 cfv 5118 (class class class)co 5767 cfn 6627 c1 7614 caddc 7616 c2 8764 ♯chash 10514 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-frec 6281 df-1o 6306 df-oadd 6310 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-inn 8714 df-2 8772 df-n0 8971 df-z 9048 df-uz 9320 df-fz 9784 df-ihash 10515 |
This theorem is referenced by: prhash2ex 10548 fiprsshashgt1 10556 |
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