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Mirrors > Home > ILE Home > Th. List > aprcl | Unicode version |
Description: Reverse closure for apartness. (Contributed by Jim Kingdon, 19-Dec-2023.) |
Ref | Expression |
---|---|
aprcl | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 3930 | . . . 4 # # | |
2 | eqeq1 2146 | . . . . . . . . . 10 | |
3 | 2 | anbi1d 460 | . . . . . . . . 9 |
4 | 3 | anbi1d 460 | . . . . . . . 8 #ℝ #ℝ #ℝ #ℝ |
5 | 4 | 2rexbidv 2460 | . . . . . . 7 #ℝ #ℝ #ℝ #ℝ |
6 | 5 | 2rexbidv 2460 | . . . . . 6 #ℝ #ℝ #ℝ #ℝ |
7 | eqeq1 2146 | . . . . . . . . . 10 | |
8 | 7 | anbi2d 459 | . . . . . . . . 9 |
9 | 8 | anbi1d 460 | . . . . . . . 8 #ℝ #ℝ #ℝ #ℝ |
10 | 9 | 2rexbidv 2460 | . . . . . . 7 #ℝ #ℝ #ℝ #ℝ |
11 | 10 | 2rexbidv 2460 | . . . . . 6 #ℝ #ℝ #ℝ #ℝ |
12 | 6, 11 | elopabi 6093 | . . . . 5 #ℝ #ℝ #ℝ #ℝ |
13 | df-ap 8347 | . . . . 5 # #ℝ #ℝ | |
14 | 12, 13 | eleq2s 2234 | . . . 4 # #ℝ #ℝ |
15 | 1, 14 | sylbi 120 | . . 3 # #ℝ #ℝ |
16 | simpl 108 | . . . . . . 7 #ℝ #ℝ | |
17 | 16 | reximi 2529 | . . . . . 6 #ℝ #ℝ |
18 | 17 | reximi 2529 | . . . . 5 #ℝ #ℝ |
19 | 18 | reximi 2529 | . . . 4 #ℝ #ℝ |
20 | 19 | reximi 2529 | . . 3 #ℝ #ℝ |
21 | 15, 20 | syl 14 | . 2 # |
22 | 13 | relopabi 4665 | . . . . . . . . . 10 # |
23 | 22 | brrelex1i 4582 | . . . . . . . . 9 # |
24 | 23 | ad3antrrr 483 | . . . . . . . 8 # |
25 | 22 | brrelex2i 4583 | . . . . . . . . 9 # |
26 | 25 | ad3antrrr 483 | . . . . . . . 8 # |
27 | op1stg 6048 | . . . . . . . 8 | |
28 | 24, 26, 27 | syl2anc 408 | . . . . . . 7 # |
29 | simprl 520 | . . . . . . . 8 # | |
30 | simprl 520 | . . . . . . . . . . 11 # | |
31 | 30 | ad2antrr 479 | . . . . . . . . . 10 # |
32 | 31 | recnd 7797 | . . . . . . . . 9 # |
33 | ax-icn 7718 | . . . . . . . . . . 11 | |
34 | 33 | a1i 9 | . . . . . . . . . 10 # |
35 | simprr 521 | . . . . . . . . . . . 12 # | |
36 | 35 | ad2antrr 479 | . . . . . . . . . . 11 # |
37 | 36 | recnd 7797 | . . . . . . . . . 10 # |
38 | 34, 37 | mulcld 7789 | . . . . . . . . 9 # |
39 | 32, 38 | addcld 7788 | . . . . . . . 8 # |
40 | 29, 39 | eqeltrd 2216 | . . . . . . 7 # |
41 | 28, 40 | eqeltrrd 2217 | . . . . . 6 # |
42 | op2ndg 6049 | . . . . . . . 8 | |
43 | 24, 26, 42 | syl2anc 408 | . . . . . . 7 # |
44 | simprr 521 | . . . . . . . 8 # | |
45 | recn 7756 | . . . . . . . . . . . 12 | |
46 | 45 | adantr 274 | . . . . . . . . . . 11 |
47 | 33 | a1i 9 | . . . . . . . . . . . 12 |
48 | recn 7756 | . . . . . . . . . . . . 13 | |
49 | 48 | adantl 275 | . . . . . . . . . . . 12 |
50 | 47, 49 | mulcld 7789 | . . . . . . . . . . 11 |
51 | 46, 50 | addcld 7788 | . . . . . . . . . 10 |
52 | 51 | adantl 275 | . . . . . . . . 9 # |
53 | 52 | adantr 274 | . . . . . . . 8 # |
54 | 44, 53 | eqeltrd 2216 | . . . . . . 7 # |
55 | 43, 54 | eqeltrrd 2217 | . . . . . 6 # |
56 | 41, 55 | jca 304 | . . . . 5 # |
57 | 56 | ex 114 | . . . 4 # |
58 | 57 | rexlimdvva 2557 | . . 3 # |
59 | 58 | rexlimdvva 2557 | . 2 # |
60 | 21, 59 | mpd 13 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wrex 2417 cvv 2686 cop 3530 class class class wbr 3929 copab 3988 cfv 5123 (class class class)co 5774 c1st 6036 c2nd 6037 cc 7621 cr 7622 ci 7625 caddc 7626 cmul 7628 #ℝ creap 8339 # cap 8346 |
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-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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-resscn 7715 ax-icn 7718 ax-addcl 7719 ax-mulcl 7721 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 df-1st 6038 df-2nd 6039 df-ap 8347 |
This theorem is referenced by: apsscn 8412 |
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