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Mirrors > Home > ILE Home > Th. List > acexmidlemcase | Unicode version |
Description: Lemma for acexmid 5531. 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 4924. 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 4272 | . . . . . . . . . . . . . 14 | |
3 | 1, 2 | eqeltri 2151 | . . . . . . . . . . . . 13 |
4 | prid1g 3496 | . . . . . . . . . . . . 13 | |
5 | 3, 4 | ax-mp 7 | . . . . . . . . . . . 12 |
6 | acexmidlem.c | . . . . . . . . . . . 12 | |
7 | 5, 6 | eleqtrri 2154 | . . . . . . . . . . 11 |
8 | eleq1 2141 | . . . . . . . . . . . . . . 15 | |
9 | 8 | anbi1d 452 | . . . . . . . . . . . . . 14 |
10 | 9 | rexbidv 2369 | . . . . . . . . . . . . 13 |
11 | 10 | reueqd 2559 | . . . . . . . . . . . 12 |
12 | 11 | rspcv 2697 | . . . . . . . . . . 11 |
13 | 7, 12 | ax-mp 7 | . . . . . . . . . 10 |
14 | riotacl 5502 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | elrabi 2746 | . . . . . . . . . 10 | |
17 | 16, 1 | eleq2s 2173 | . . . . . . . . 9 |
18 | elpri 3421 | . . . . . . . . 9 | |
19 | 15, 17, 18 | 3syl 17 | . . . . . . . 8 |
20 | eleq1 2141 | . . . . . . . . . 10 | |
21 | 15, 20 | syl5ibcom 153 | . . . . . . . . 9 |
22 | 21 | orim2d 734 | . . . . . . . 8 |
23 | 19, 22 | mpd 13 | . . . . . . 7 |
24 | acexmidlem.b | . . . . . . . . . . . . . 14 | |
25 | pp0ex 3960 | . . . . . . . . . . . . . . 15 | |
26 | 25 | rabex 3922 | . . . . . . . . . . . . . 14 |
27 | 24, 26 | eqeltri 2151 | . . . . . . . . . . . . 13 |
28 | 27 | prid2 3499 | . . . . . . . . . . . 12 |
29 | 28, 6 | eleqtrri 2154 | . . . . . . . . . . 11 |
30 | eleq1 2141 | . . . . . . . . . . . . . . 15 | |
31 | 30 | anbi1d 452 | . . . . . . . . . . . . . 14 |
32 | 31 | rexbidv 2369 | . . . . . . . . . . . . 13 |
33 | 32 | reueqd 2559 | . . . . . . . . . . . 12 |
34 | 33 | rspcv 2697 | . . . . . . . . . . 11 |
35 | 29, 34 | ax-mp 7 | . . . . . . . . . 10 |
36 | riotacl 5502 | . . . . . . . . . 10 | |
37 | 35, 36 | syl 14 | . . . . . . . . 9 |
38 | elrabi 2746 | . . . . . . . . . 10 | |
39 | 38, 24 | eleq2s 2173 | . . . . . . . . 9 |
40 | elpri 3421 | . . . . . . . . 9 | |
41 | 37, 39, 40 | 3syl 17 | . . . . . . . 8 |
42 | eleq1 2141 | . . . . . . . . . 10 | |
43 | 37, 42 | syl5ibcom 153 | . . . . . . . . 9 |
44 | 43 | orim1d 733 | . . . . . . . 8 |
45 | 41, 44 | mpd 13 | . . . . . . 7 |
46 | 23, 45 | jca 300 | . . . . . 6 |
47 | anddi 767 | . . . . . 6 | |
48 | 46, 47 | sylib 120 | . . . . 5 |
49 | simpl 107 | . . . . . . 7 | |
50 | simpl 107 | . . . . . . 7 | |
51 | 49, 50 | jaoi 668 | . . . . . 6 |
52 | 51 | orim2i 710 | . . . . 5 |
53 | 48, 52 | syl 14 | . . . 4 |
54 | 53 | orcomd 680 | . . 3 |
55 | simpr 108 | . . . . 5 | |
56 | 55 | orim1i 709 | . . . 4 |
57 | 56 | orim2i 710 | . . 3 |
58 | 54, 57 | syl 14 | . 2 |
59 | 3orass 922 | . 2 | |
60 | 58, 59 | sylibr 132 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 w3o 918 wceq 1284 wcel 1433 wral 2348 wrex 2349 wreu 2350 crab 2352 cvv 2601 c0 3251 csn 3398 cpr 3399 con0 4118 crio 5487 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-tr 3876 df-iord 4121 df-on 4123 df-suc 4126 df-iota 4887 df-riota 5488 |
This theorem is referenced by: acexmidlem1 5528 |
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