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Mirrors > Home > ILE Home > Th. List > acexmidlemcase | Unicode version |
Description: Lemma for acexmid 5766. Here we divide the proof into cases (based
on the
disjunction implicit in an unordered pair, not the sort of case
elimination which relies on excluded middle).
The cases are (1) the choice function evaluated at equals , (2) the choice function evaluated at equals , and (3) the choice function evaluated at equals and the choice function evaluated at equals . Because of the way we represent the choice function , the choice function evaluated at is and the choice function evaluated at is . Other than the difference in notation these work just as and would if were a function as defined by df-fun 5120. Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at equals , then and likewise for . (Contributed by Jim Kingdon, 7-Aug-2019.) |
Ref | Expression |
---|---|
acexmidlem.a | |
acexmidlem.b | |
acexmidlem.c |
Ref | Expression |
---|---|
acexmidlemcase |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acexmidlem.a | . . . . . . . . . . . . . 14 | |
2 | onsucelsucexmidlem 4439 | . . . . . . . . . . . . . 14 | |
3 | 1, 2 | eqeltri 2210 | . . . . . . . . . . . . 13 |
4 | prid1g 3622 | . . . . . . . . . . . . 13 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . . 12 |
6 | acexmidlem.c | . . . . . . . . . . . 12 | |
7 | 5, 6 | eleqtrri 2213 | . . . . . . . . . . 11 |
8 | eleq1 2200 | . . . . . . . . . . . . . . 15 | |
9 | 8 | anbi1d 460 | . . . . . . . . . . . . . 14 |
10 | 9 | rexbidv 2436 | . . . . . . . . . . . . 13 |
11 | 10 | reueqd 2634 | . . . . . . . . . . . 12 |
12 | 11 | rspcv 2780 | . . . . . . . . . . 11 |
13 | 7, 12 | ax-mp 5 | . . . . . . . . . 10 |
14 | riotacl 5737 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | elrabi 2832 | . . . . . . . . . 10 | |
17 | 16, 1 | eleq2s 2232 | . . . . . . . . 9 |
18 | elpri 3545 | . . . . . . . . 9 | |
19 | 15, 17, 18 | 3syl 17 | . . . . . . . 8 |
20 | eleq1 2200 | . . . . . . . . . 10 | |
21 | 15, 20 | syl5ibcom 154 | . . . . . . . . 9 |
22 | 21 | orim2d 777 | . . . . . . . 8 |
23 | 19, 22 | mpd 13 | . . . . . . 7 |
24 | acexmidlem.b | . . . . . . . . . . . . . 14 | |
25 | pp0ex 4108 | . . . . . . . . . . . . . . 15 | |
26 | 25 | rabex 4067 | . . . . . . . . . . . . . 14 |
27 | 24, 26 | eqeltri 2210 | . . . . . . . . . . . . 13 |
28 | 27 | prid2 3625 | . . . . . . . . . . . 12 |
29 | 28, 6 | eleqtrri 2213 | . . . . . . . . . . 11 |
30 | eleq1 2200 | . . . . . . . . . . . . . . 15 | |
31 | 30 | anbi1d 460 | . . . . . . . . . . . . . 14 |
32 | 31 | rexbidv 2436 | . . . . . . . . . . . . 13 |
33 | 32 | reueqd 2634 | . . . . . . . . . . . 12 |
34 | 33 | rspcv 2780 | . . . . . . . . . . 11 |
35 | 29, 34 | ax-mp 5 | . . . . . . . . . 10 |
36 | riotacl 5737 | . . . . . . . . . 10 | |
37 | 35, 36 | syl 14 | . . . . . . . . 9 |
38 | elrabi 2832 | . . . . . . . . . 10 | |
39 | 38, 24 | eleq2s 2232 | . . . . . . . . 9 |
40 | elpri 3545 | . . . . . . . . 9 | |
41 | 37, 39, 40 | 3syl 17 | . . . . . . . 8 |
42 | eleq1 2200 | . . . . . . . . . 10 | |
43 | 37, 42 | syl5ibcom 154 | . . . . . . . . 9 |
44 | 43 | orim1d 776 | . . . . . . . 8 |
45 | 41, 44 | mpd 13 | . . . . . . 7 |
46 | 23, 45 | jca 304 | . . . . . 6 |
47 | anddi 810 | . . . . . 6 | |
48 | 46, 47 | sylib 121 | . . . . 5 |
49 | simpl 108 | . . . . . . 7 | |
50 | simpl 108 | . . . . . . 7 | |
51 | 49, 50 | jaoi 705 | . . . . . 6 |
52 | 51 | orim2i 750 | . . . . 5 |
53 | 48, 52 | syl 14 | . . . 4 |
54 | 53 | orcomd 718 | . . 3 |
55 | simpr 109 | . . . . 5 | |
56 | 55 | orim1i 749 | . . . 4 |
57 | 56 | orim2i 750 | . . 3 |
58 | 54, 57 | syl 14 | . 2 |
59 | 3orass 965 | . 2 | |
60 | 58, 59 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3o 961 wceq 1331 wcel 1480 wral 2414 wrex 2415 wreu 2416 crab 2418 cvv 2681 c0 3358 csn 3522 cpr 3523 con0 4280 crio 5722 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 df-iota 5083 df-riota 5723 |
This theorem is referenced by: acexmidlem1 5763 |
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