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Mirrors > Home > ILE Home > Th. List > acexmidlemcase | Unicode version |
Description: Lemma for acexmid 5852. 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 5200. 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 4513 | . . . . . . . . . . . . . 14 | |
3 | 1, 2 | eqeltri 2243 | . . . . . . . . . . . . 13 |
4 | prid1g 3687 | . . . . . . . . . . . . 13 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . . 12 |
6 | acexmidlem.c | . . . . . . . . . . . 12 | |
7 | 5, 6 | eleqtrri 2246 | . . . . . . . . . . 11 |
8 | eleq1 2233 | . . . . . . . . . . . . . . 15 | |
9 | 8 | anbi1d 462 | . . . . . . . . . . . . . 14 |
10 | 9 | rexbidv 2471 | . . . . . . . . . . . . 13 |
11 | 10 | reueqd 2675 | . . . . . . . . . . . 12 |
12 | 11 | rspcv 2830 | . . . . . . . . . . 11 |
13 | 7, 12 | ax-mp 5 | . . . . . . . . . 10 |
14 | riotacl 5823 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | elrabi 2883 | . . . . . . . . . 10 | |
17 | 16, 1 | eleq2s 2265 | . . . . . . . . 9 |
18 | elpri 3606 | . . . . . . . . 9 | |
19 | 15, 17, 18 | 3syl 17 | . . . . . . . 8 |
20 | eleq1 2233 | . . . . . . . . . 10 | |
21 | 15, 20 | syl5ibcom 154 | . . . . . . . . 9 |
22 | 21 | orim2d 783 | . . . . . . . 8 |
23 | 19, 22 | mpd 13 | . . . . . . 7 |
24 | acexmidlem.b | . . . . . . . . . . . . . 14 | |
25 | pp0ex 4175 | . . . . . . . . . . . . . . 15 | |
26 | 25 | rabex 4133 | . . . . . . . . . . . . . 14 |
27 | 24, 26 | eqeltri 2243 | . . . . . . . . . . . . 13 |
28 | 27 | prid2 3690 | . . . . . . . . . . . 12 |
29 | 28, 6 | eleqtrri 2246 | . . . . . . . . . . 11 |
30 | eleq1 2233 | . . . . . . . . . . . . . . 15 | |
31 | 30 | anbi1d 462 | . . . . . . . . . . . . . 14 |
32 | 31 | rexbidv 2471 | . . . . . . . . . . . . 13 |
33 | 32 | reueqd 2675 | . . . . . . . . . . . 12 |
34 | 33 | rspcv 2830 | . . . . . . . . . . 11 |
35 | 29, 34 | ax-mp 5 | . . . . . . . . . 10 |
36 | riotacl 5823 | . . . . . . . . . 10 | |
37 | 35, 36 | syl 14 | . . . . . . . . 9 |
38 | elrabi 2883 | . . . . . . . . . 10 | |
39 | 38, 24 | eleq2s 2265 | . . . . . . . . 9 |
40 | elpri 3606 | . . . . . . . . 9 | |
41 | 37, 39, 40 | 3syl 17 | . . . . . . . 8 |
42 | eleq1 2233 | . . . . . . . . . 10 | |
43 | 37, 42 | syl5ibcom 154 | . . . . . . . . 9 |
44 | 43 | orim1d 782 | . . . . . . . 8 |
45 | 41, 44 | mpd 13 | . . . . . . 7 |
46 | 23, 45 | jca 304 | . . . . . 6 |
47 | anddi 816 | . . . . . 6 | |
48 | 46, 47 | sylib 121 | . . . . 5 |
49 | simpl 108 | . . . . . . 7 | |
50 | simpl 108 | . . . . . . 7 | |
51 | 49, 50 | jaoi 711 | . . . . . 6 |
52 | 51 | orim2i 756 | . . . . 5 |
53 | 48, 52 | syl 14 | . . . 4 |
54 | 53 | orcomd 724 | . . 3 |
55 | simpr 109 | . . . . 5 | |
56 | 55 | orim1i 755 | . . . 4 |
57 | 56 | orim2i 756 | . . 3 |
58 | 54, 57 | syl 14 | . 2 |
59 | 3orass 976 | . 2 | |
60 | 58, 59 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 703 w3o 972 wceq 1348 wcel 2141 wral 2448 wrex 2449 wreu 2450 crab 2452 cvv 2730 c0 3414 csn 3583 cpr 3584 con0 4348 crio 5808 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-tr 4088 df-iord 4351 df-on 4353 df-suc 4356 df-iota 5160 df-riota 5809 |
This theorem is referenced by: acexmidlem1 5849 |
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