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Mirrors > Home > ILE Home > Th. List > exmidaclem | Unicode version |
Description: Lemma for exmidac 7065. The result, with a few hypotheses to break out commonly used expressions. (Contributed by Jim Kingdon, 21-Nov-2023.) |
Ref | Expression |
---|---|
exmidaclem.a | |
exmidaclem.b | |
exmidaclem.c |
Ref | Expression |
---|---|
exmidaclem | CHOICE EXMID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 108 | . . . 4 CHOICE CHOICE | |
2 | exmidaclem.c | . . . . . 6 | |
3 | exmidaclem.a | . . . . . . . 8 | |
4 | pp0ex 4113 | . . . . . . . . 9 | |
5 | 4 | rabex 4072 | . . . . . . . 8 |
6 | 3, 5 | eqeltri 2212 | . . . . . . 7 |
7 | exmidaclem.b | . . . . . . . 8 | |
8 | 4 | rabex 4072 | . . . . . . . 8 |
9 | 7, 8 | eqeltri 2212 | . . . . . . 7 |
10 | prexg 4133 | . . . . . . 7 | |
11 | 6, 9, 10 | mp2an 422 | . . . . . 6 |
12 | 2, 11 | eqeltri 2212 | . . . . 5 |
13 | 12 | a1i 9 | . . . 4 CHOICE |
14 | simpr 109 | . . . . . . 7 CHOICE | |
15 | 14, 2 | eleqtrdi 2232 | . . . . . 6 CHOICE |
16 | elpri 3550 | . . . . . 6 | |
17 | 0ex 4055 | . . . . . . . . . . 11 | |
18 | 17 | prid1 3629 | . . . . . . . . . 10 |
19 | eqid 2139 | . . . . . . . . . . 11 | |
20 | 19 | orci 720 | . . . . . . . . . 10 |
21 | eqeq1 2146 | . . . . . . . . . . . 12 | |
22 | 21 | orbi1d 780 | . . . . . . . . . . 11 |
23 | 22, 3 | elrab2 2843 | . . . . . . . . . 10 |
24 | 18, 20, 23 | mpbir2an 926 | . . . . . . . . 9 |
25 | eleq2 2203 | . . . . . . . . 9 | |
26 | 24, 25 | mpbiri 167 | . . . . . . . 8 |
27 | elex2 2702 | . . . . . . . 8 | |
28 | 26, 27 | syl 14 | . . . . . . 7 |
29 | p0ex 4112 | . . . . . . . . . . 11 | |
30 | 29 | prid2 3630 | . . . . . . . . . 10 |
31 | eqid 2139 | . . . . . . . . . . 11 | |
32 | 31 | orci 720 | . . . . . . . . . 10 |
33 | eqeq1 2146 | . . . . . . . . . . . 12 | |
34 | 33 | orbi1d 780 | . . . . . . . . . . 11 |
35 | 34, 7 | elrab2 2843 | . . . . . . . . . 10 |
36 | 30, 32, 35 | mpbir2an 926 | . . . . . . . . 9 |
37 | eleq2 2203 | . . . . . . . . 9 | |
38 | 36, 37 | mpbiri 167 | . . . . . . . 8 |
39 | elex2 2702 | . . . . . . . 8 | |
40 | 38, 39 | syl 14 | . . . . . . 7 |
41 | 28, 40 | jaoi 705 | . . . . . 6 |
42 | 15, 16, 41 | 3syl 17 | . . . . 5 CHOICE |
43 | 42 | ralrimiva 2505 | . . . 4 CHOICE |
44 | 1, 13, 43 | acfun 7063 | . . 3 CHOICE |
45 | 0nep0 4089 | . . . . . . . . . 10 | |
46 | 45 | neii 2310 | . . . . . . . . 9 |
47 | simplr 519 | . . . . . . . . . 10 CHOICE | |
48 | simpr 109 | . . . . . . . . . 10 CHOICE | |
49 | 47, 48 | eqeq12d 2154 | . . . . . . . . 9 CHOICE |
50 | 46, 49 | mtbiri 664 | . . . . . . . 8 CHOICE |
51 | olc 700 | . . . . . . . . . . . . 13 | |
52 | 51 | ralrimivw 2506 | . . . . . . . . . . . 12 |
53 | rabid2 2607 | . . . . . . . . . . . 12 | |
54 | 52, 53 | sylibr 133 | . . . . . . . . . . 11 |
55 | 54, 3 | syl6eqr 2190 | . . . . . . . . . 10 |
56 | olc 700 | . . . . . . . . . . . . 13 | |
57 | 56 | ralrimivw 2506 | . . . . . . . . . . . 12 |
58 | rabid2 2607 | . . . . . . . . . . . 12 | |
59 | 57, 58 | sylibr 133 | . . . . . . . . . . 11 |
60 | 59, 7 | syl6eqr 2190 | . . . . . . . . . 10 |
61 | 55, 60 | eqtr3d 2174 | . . . . . . . . 9 |
62 | 61 | fveq2d 5425 | . . . . . . . 8 |
63 | 50, 62 | nsyl 617 | . . . . . . 7 CHOICE |
64 | 63 | olcd 723 | . . . . . 6 CHOICE |
65 | simpr 109 | . . . . . . 7 CHOICE | |
66 | 65 | orcd 722 | . . . . . 6 CHOICE |
67 | fveq2 5421 | . . . . . . . . . . 11 | |
68 | id 19 | . . . . . . . . . . 11 | |
69 | 67, 68 | eleq12d 2210 | . . . . . . . . . 10 |
70 | simprr 521 | . . . . . . . . . 10 CHOICE | |
71 | 9 | prid2 3630 | . . . . . . . . . . . 12 |
72 | 71, 2 | eleqtrri 2215 | . . . . . . . . . . 11 |
73 | 72 | a1i 9 | . . . . . . . . . 10 CHOICE |
74 | 69, 70, 73 | rspcdva 2794 | . . . . . . . . 9 CHOICE |
75 | eqeq1 2146 | . . . . . . . . . . 11 | |
76 | 75 | orbi1d 780 | . . . . . . . . . 10 |
77 | 76, 7 | elrab2 2843 | . . . . . . . . 9 |
78 | 74, 77 | sylib 121 | . . . . . . . 8 CHOICE |
79 | 78 | simprd 113 | . . . . . . 7 CHOICE |
80 | 79 | adantr 274 | . . . . . 6 CHOICE |
81 | 64, 66, 80 | mpjaodan 787 | . . . . 5 CHOICE |
82 | df-dc 820 | . . . . 5 DECID | |
83 | 81, 82 | sylibr 133 | . . . 4 CHOICE DECID |
84 | simpr 109 | . . . . . 6 CHOICE | |
85 | 84 | orcd 722 | . . . . 5 CHOICE |
86 | 85, 82 | sylibr 133 | . . . 4 CHOICE DECID |
87 | fveq2 5421 | . . . . . . . 8 | |
88 | id 19 | . . . . . . . 8 | |
89 | 87, 88 | eleq12d 2210 | . . . . . . 7 |
90 | 6 | prid1 3629 | . . . . . . . . 9 |
91 | 90, 2 | eleqtrri 2215 | . . . . . . . 8 |
92 | 91 | a1i 9 | . . . . . . 7 CHOICE |
93 | 89, 70, 92 | rspcdva 2794 | . . . . . 6 CHOICE |
94 | eqeq1 2146 | . . . . . . . 8 | |
95 | 94 | orbi1d 780 | . . . . . . 7 |
96 | 95, 3 | elrab2 2843 | . . . . . 6 |
97 | 93, 96 | sylib 121 | . . . . 5 CHOICE |
98 | 97 | simprd 113 | . . . 4 CHOICE |
99 | 83, 86, 98 | mpjaodan 787 | . . 3 CHOICE DECID |
100 | 44, 99 | exlimddv 1870 | . 2 CHOICE DECID |
101 | 100 | exmid1dc 4123 | 1 CHOICE EXMID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wex 1468 wcel 1480 wral 2416 crab 2420 cvv 2686 wss 3071 c0 3363 csn 3527 cpr 3528 EXMIDwem 4118 wfn 5118 cfv 5123 CHOICEwac 7061 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 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-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-exmid 4119 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ac 7062 |
This theorem is referenced by: exmidac 7065 |
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