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Mirrors > Home > ILE Home > Th. List > acexmidlemcase | Unicode version |
Description: Lemma for acexmid 5824. 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 5173. 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 4489 | . . . . . . . . . . . . . 14 | |
3 | 1, 2 | eqeltri 2230 | . . . . . . . . . . . . 13 |
4 | prid1g 3664 | . . . . . . . . . . . . 13 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . . 12 |
6 | acexmidlem.c | . . . . . . . . . . . 12 | |
7 | 5, 6 | eleqtrri 2233 | . . . . . . . . . . 11 |
8 | eleq1 2220 | . . . . . . . . . . . . . . 15 | |
9 | 8 | anbi1d 461 | . . . . . . . . . . . . . 14 |
10 | 9 | rexbidv 2458 | . . . . . . . . . . . . 13 |
11 | 10 | reueqd 2662 | . . . . . . . . . . . 12 |
12 | 11 | rspcv 2812 | . . . . . . . . . . 11 |
13 | 7, 12 | ax-mp 5 | . . . . . . . . . 10 |
14 | riotacl 5795 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | elrabi 2865 | . . . . . . . . . 10 | |
17 | 16, 1 | eleq2s 2252 | . . . . . . . . 9 |
18 | elpri 3583 | . . . . . . . . 9 | |
19 | 15, 17, 18 | 3syl 17 | . . . . . . . 8 |
20 | eleq1 2220 | . . . . . . . . . 10 | |
21 | 15, 20 | syl5ibcom 154 | . . . . . . . . 9 |
22 | 21 | orim2d 778 | . . . . . . . 8 |
23 | 19, 22 | mpd 13 | . . . . . . 7 |
24 | acexmidlem.b | . . . . . . . . . . . . . 14 | |
25 | pp0ex 4151 | . . . . . . . . . . . . . . 15 | |
26 | 25 | rabex 4109 | . . . . . . . . . . . . . 14 |
27 | 24, 26 | eqeltri 2230 | . . . . . . . . . . . . 13 |
28 | 27 | prid2 3667 | . . . . . . . . . . . 12 |
29 | 28, 6 | eleqtrri 2233 | . . . . . . . . . . 11 |
30 | eleq1 2220 | . . . . . . . . . . . . . . 15 | |
31 | 30 | anbi1d 461 | . . . . . . . . . . . . . 14 |
32 | 31 | rexbidv 2458 | . . . . . . . . . . . . 13 |
33 | 32 | reueqd 2662 | . . . . . . . . . . . 12 |
34 | 33 | rspcv 2812 | . . . . . . . . . . 11 |
35 | 29, 34 | ax-mp 5 | . . . . . . . . . 10 |
36 | riotacl 5795 | . . . . . . . . . 10 | |
37 | 35, 36 | syl 14 | . . . . . . . . 9 |
38 | elrabi 2865 | . . . . . . . . . 10 | |
39 | 38, 24 | eleq2s 2252 | . . . . . . . . 9 |
40 | elpri 3583 | . . . . . . . . 9 | |
41 | 37, 39, 40 | 3syl 17 | . . . . . . . 8 |
42 | eleq1 2220 | . . . . . . . . . 10 | |
43 | 37, 42 | syl5ibcom 154 | . . . . . . . . 9 |
44 | 43 | orim1d 777 | . . . . . . . 8 |
45 | 41, 44 | mpd 13 | . . . . . . 7 |
46 | 23, 45 | jca 304 | . . . . . 6 |
47 | anddi 811 | . . . . . 6 | |
48 | 46, 47 | sylib 121 | . . . . 5 |
49 | simpl 108 | . . . . . . 7 | |
50 | simpl 108 | . . . . . . 7 | |
51 | 49, 50 | jaoi 706 | . . . . . 6 |
52 | 51 | orim2i 751 | . . . . 5 |
53 | 48, 52 | syl 14 | . . . 4 |
54 | 53 | orcomd 719 | . . 3 |
55 | simpr 109 | . . . . 5 | |
56 | 55 | orim1i 750 | . . . 4 |
57 | 56 | orim2i 751 | . . 3 |
58 | 54, 57 | syl 14 | . 2 |
59 | 3orass 966 | . 2 | |
60 | 58, 59 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 w3o 962 wceq 1335 wcel 2128 wral 2435 wrex 2436 wreu 2437 crab 2439 cvv 2712 c0 3394 csn 3560 cpr 3561 con0 4324 crio 5780 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-uni 3774 df-tr 4064 df-iord 4327 df-on 4329 df-suc 4332 df-iota 5136 df-riota 5781 |
This theorem is referenced by: acexmidlem1 5821 |
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